#7632616

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

5054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=hi,e^2=gi,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, 7539812);