#7726304

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	#51531	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4578 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7633500);