#7630569

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5032 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)(9,10,13,16)(11,12,15,14)(17,18,23,24)(19,26,20,21)(22,30,31,25)(27,29,28,32);(9,11,13,15)(10,14,16,12)(17,19,23,20)(18,21,24,26)(22,27,31,28)(25,29,30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(18,24)(19,20)(25,30)(27,28);(17,20,23,19)(18,21,24,26)(22,28,31,27)(25,29,30,32);(17,21)(18,19)(20,24)(22,29)(23,26)(25,27)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(17,22,23,31)(18,25,24,30)(19,27,20,28)(21,29,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537765);