#6516940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(9,10)(11,19,22,32)(12,17)(13,18)(14,16)(15,21,27,25)(20,33,34,40)(23,29)(24,31)(26,30)(28,35,38,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,25)(14,22)(16,27)(17,28)(18,32)(19,30)(21,26)(23,34)(24,37)(29,38)(31,40)(33,39)(35,36);(2,5)(3,7)(10,16)(13,26)(15,27)(17,29)(21,25)(24,36)(28,38)(35,37);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,28)(16,29)(18,31)(19,33)(21,35)(22,34)(25,37)(26,36)(27,38)(30,39)(32,40);(2,5)(6,8)(12,23)(17,29)(20,34)(24,36)(28,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319333
maximal normal subgroup of	#72319333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424136);