#7640638

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

4320 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)(9,10,14,11)(12,17,20,15)(13,18,21,16)(19,22,24,23)(25,26)(27,30)(28,35)(29,32)(31,37)(33,40)(34,39)(36,38);(2,5)(6,8)(9,11,14,10)(12,17,20,15)(13,16,21,18)(19,22,24,23)(25,27,29,34)(26,30,32,39)(28,36,37,33)(31,40,35,38);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(26,32)(30,39)(31,35)(38,40);(12,20)(15,17)(19,24)(22,23);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi

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, 7547834);