#8093073

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000269);