#495574

order	512
nilpotency class	4
isoclinism	isoclinic class of #495573
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#21408
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3060	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=h,d^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=dh,f^b=fi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240824, #72249754, #72249766, …
maximal normal subgroup of	#72240824, #72249754, #72249766, …
maximal quotient of	#72240739, #72250035, #72250126, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402770);