#7918402

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4558 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825598);