#232508

order	512
nilpotency class	3
isoclinism	isoclinic class of #9980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#392
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303845
maximal normal subgroup of	#72303845
maximal quotient of	#72304010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139704);