#8926043

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	#36833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833239);