#7927130

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=hi,f^2=i,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, 7834326);