#7712122

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4794 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7619318);