#7631713

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
prime radical	#35807	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4932 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)(10,14)(15,16)(17,18,23,22)(19,27,30,24)(20,25,29,31)(21,26,28,32);(1,2)(3,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,28)(21,29)(22,27)(23,30)(25,26)(31,32);(1,3,6,8)(2,4,7,5)(17,20)(18,25)(19,21)(22,31)(23,29)(24,32)(26,27)(28,30);(1,4,6,5)(2,3,7,8)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,26)(27,32);(1,4,6,5)(2,3,7,8)(9,11,13,12)(10,15,14,16)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,25,27,31);(1,5,6,4)(2,8,7,3)(9,12,13,11)(10,16,14,15)(17,22,23,18)(19,24,30,27)(20,31,29,25)(21,32,28,26)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=g,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, 7538909);