#8751700

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658896);