#8456574

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

5516 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342290, #72342326, #72342345
maximal normal subgroup of	#72342290, #72342326, #72342345

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8363770);