#8445776

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

4817 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)(10,13)(14,16)(17,18,24,28)(19,25,22,23)(20,30,29,26)(21,27,31,32);(2,3)(6,7)(11,15)(14,16)(17,19,24,22)(18,25,28,23)(20,31,29,21)(26,27,30,32);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,26)(19,21)(22,31)(23,27)(24,29)(25,32)(28,30);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,24,31)(18,27,28,32)(19,29,22,20)(23,30,25,26);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)(17,22,24,19)(18,23,28,25)(20,31,29,21)(26,27,30,32);(17,23)(18,19)(20,27)(21,26)(22,28)(24,25)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=h,e^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352972);