#376349

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

905 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)(3,5)(4,6)(7,8)(9,10,15,20,16,21,14,32)(11,22,25,17,27,36,26,37)(12,18,30,39,24,35,29,40)(13,31,23,19,28,34,33,38);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,28)(13,24)(14,26)(15,25)(16,27)(18,38)(19,35)(20,36)(21,37)(22,32)(23,29)(30,33)(31,39)(34,40);(2,6)(5,8)(9,12)(10,18)(11,23)(13,26)(14,29)(15,30)(16,24)(17,34)(19,22)(20,39)(21,35)(25,28)(27,33)(31,37)(32,40)(36,38);(9,13,15,23,16,28,14,33)(10,19,20,34,21,38,32,31)(11,24,25,29,27,12,26,30)(17,35,36,40,37,18,22,39)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,e^2=i,g^2=i,h^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, 283545);