#7682528

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4552 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^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, 7589724);