#361704

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,g^c=gi,e^d=ei

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, 268900);