#7629674

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4952 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,c^b=ci,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536870);