#8699281

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262811, #72311356, #72311396, …
maximal normal subgroup of	#72262811, #72311356, #72311396, …
maximal quotient of	#72262822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8606477);