#496806

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22030
prime radical	#3025	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

638 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=ceh,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224962, #72343187
maximal normal subgroup of	#72224962, #72343187

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404002);