#8422271

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	#37484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8329467);