#7678425

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4692 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,e^2=hi,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7585621);