#7630846

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	#35771	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538042);