#7632044

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

5322 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539240);