#7743212

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	#52013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4800 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650408);