#7702392

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#45788	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(2,5)(6,8)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(10,15)(12,19)(14,20)(17,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(9,12,13,19)(10,16,15,22)(11,17,18,23)(14,21,20,24)(26,32)(28,36)(30,38)(33,35);(10,15)(14,20)(16,22)(21,24)(28,36)(31,40)(33,35)(37,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,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, 7609588);