#376763

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1309 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=h,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^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283959);