#7734349

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	#43019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4998 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)(9,10,13,14)(11,12,15,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,30)(31,32);(1,3,5,8)(2,6,4,7)(17,19,23,21)(18,24,27,28)(20,26,30,31)(22,25,32,29);(10,14)(11,15)(19,21)(20,30)(24,28)(25,29);(9,11)(10,12)(13,15)(14,16)(17,20)(18,25)(19,26)(21,31)(22,28)(23,30)(24,32)(27,29);(1,2,5,4)(3,6,8,7)(9,12)(10,11)(13,16)(14,15)(17,21,23,19)(18,24,27,28)(20,31,30,26)(22,25,32,29);(1,4,5,2)(3,7,8,6)(17,22,23,32)(18,26,27,31)(19,29,21,25)(20,28,30,24)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641545);