#8093052

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4593 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)(3,5)(4,7)(6,8)(9,10)(11,18)(12,16)(13,15)(14,17)(19,24)(20,23)(21,22)(25,26)(27,31)(28,32)(29,37)(30,33)(34,38)(35,39)(36,40);(1,3,5,2)(4,7,8,6)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,27,30,29)(26,31,33,37)(28,35,36,34)(32,39,40,38);(1,4,5,8)(2,6,3,7)(9,12,14,20)(10,16,17,23)(11,19,13,21)(15,22,18,24)(25,28)(26,32)(27,34)(29,35)(30,36)(31,38)(33,40)(37,39);(2,3)(6,7)(10,17)(12,20)(15,18)(19,21)(28,36)(32,40)(34,35)(38,39);(10,17)(15,18)(16,23)(22,24)(25,29,30,27)(26,31,33,37)(28,35,36,34)(32,38,40,39);(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,e^2=i,f^2=gh,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, 8000248);