#378493

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 285689);