#7701812

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23);(2,5)(6,8)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(11,18)(14,17)(19,24)(22,23)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(27,35)(30,33)(34,40)(38,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,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, 7609008);