#361701

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

3516 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)(18,29)(21,24)(22,33)(25,31)(26,32)(30,34)(35,37)(36,39)(38,40);(9,11,16,23,15,24,14,31)(10,17,19,21,12,25,27,20)(13,22,29,39,30,38,32,37)(18,33,34,36,26,40,28,35);(2,4)(9,12,15,10)(11,21,24,20)(13,26,30,18)(14,27,16,19)(17,31,25,23)(22,36,38,35)(28,29,34,32)(33,37,40,39);(9,13)(10,18)(11,22)(12,26)(14,32)(15,30)(16,29)(17,33)(19,34)(20,35)(21,36)(23,39)(24,38)(25,40)(27,28)(31,37)

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,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268897);