#6458272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2487
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2,8,12,7,13,6,30)(3,14,18,9,17,34,19,33)(4,20,23,37,25,36,24,10)(5,11,27,39,29,40,28,38)(15,31,46,48,21,50,44,41)(16,42,47,32,26,55,45,57)(22,49,54,58,53,59,52,35)(43,56,61,63,51,64,62,60);(1,3)(2,9)(4,21)(5,26)(6,19)(7,17)(8,18)(10,31)(11,32)(12,34)(13,33)(14,30)(15,25)(16,29)(20,48)(22,43)(23,44)(24,46)(27,45)(28,47)(35,56)(36,41)(37,50)(38,42)(39,55)(40,57)(49,63)(51,53)(52,62)(54,61)(58,64)(59,60);(1,4,6,24,7,25,8,23)(2,10,12,20,13,37,30,36)(3,15,19,44,17,21,18,46)(5,22,28,52,29,53,27,54)(9,31,34,48,33,50,14,41)(11,35,39,49,40,58,38,59)(16,43,45,62,26,51,47,61)(32,56,55,63,57,64,42,60);(1,5)(2,11)(3,16)(4,22)(6,28)(7,29)(8,27)(9,32)(10,35)(12,39)(13,40)(14,42)(15,43)(17,26)(18,47)(19,45)(20,49)(21,51)(23,54)(24,52)(25,53)(30,38)(31,56)(33,57)(34,55)(36,59)(37,58)(41,60)(44,62)(46,61)(48,63)(50,64);(3,17)(9,33)(14,34)(15,21)(16,26)(18,19)(31,50)(32,57)(41,48)(42,55)(43,51)(44,46)(45,47)(56,64)(60,63)(61,62)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249776, #72249781, #72249829, …
maximal normal subgroup of	#72249776, #72249781, #72249829, …
maximal quotient of	#72256961, #72322778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365468);