#7682534

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	#43580	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4557 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)(14,15)(17,18,23,27)(19,28,29,24)(20,21,25,32)(22,31,30,26);(1,2)(3,6)(4,5)(7,8)(17,19,23,29)(18,24,27,28)(20,30,25,22)(21,31,32,26);(9,10,12,13)(11,15,16,14)(19,29)(22,30)(24,28)(26,31);(9,11,12,16)(10,14,13,15)(17,20)(18,21)(19,30)(22,29)(23,25)(24,31)(26,28)(27,32);(17,21)(18,25)(19,26)(20,27)(22,28)(23,32)(24,30)(29,31);(1,3,4,7)(2,6,5,8)(17,22,23,30)(18,26,27,31)(19,20,29,25)(21,28,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589730);