#384859

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	#2361
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1051 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)(9,10,14,22,15,21,16,20)(11,23,27,38,26,30,25,17)(12,28,31,39,32,18,37,24)(13,33,34,40,35,29,36,19);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,29)(13,18)(14,27)(15,26)(16,25)(19,31)(20,30)(21,38)(22,23)(24,34)(28,35)(32,33)(36,39)(37,40);(2,6)(5,7)(9,12)(10,18)(11,19)(13,30)(14,31)(15,32)(16,37)(17,34)(20,39)(21,28)(22,24)(23,35)(25,29)(26,40)(27,33)(36,38);(3,8)(5,7)(9,13,16,36,15,35,14,34)(10,19,22,33,21,40,20,29)(11,24,27,28,26,39,25,18)(12,30,37,38,32,23,31,17)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292055);