#10323034

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	#57995	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 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:

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

Polycyclic presentation:

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

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, 10230230);