#372341

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
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	D₄×C₂²⋊C₄
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(3,6)(4,5)(7,8)(9,10,15,20,16,22,14,21)(11,23,24,17,26,35,25,36)(12,27,30,18,31,38,29,39)(13,32,28,19,34,40,33,37);(9,11,15,24,16,26,14,25)(10,17,20,35,22,36,21,23)(12,28,30,34,31,33,29,13)(18,37,38,32,39,19,27,40);(2,5)(6,8)(9,12)(10,18)(11,13)(14,29)(15,30)(16,31)(17,32)(19,35)(20,38)(21,27)(22,39)(23,37)(24,28)(25,33)(26,34)(36,40);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,12)(14,33)(15,28)(16,34)(17,27)(18,35)(20,40)(21,32)(22,37)(23,39)(24,30)(25,29)(26,31)(36,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=fg,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 279537);