#372623

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1097 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)(4,5)(6,8)(9,10,15,20,16,22,14,21)(11,23,25,17,12,18,26,33)(13,28,29,19,30,36,31,35)(24,32,39,34,27,40,38,37);(2,5)(6,7)(9,11,15,25,16,12,14,26)(10,17,20,18,22,33,21,23)(13,24,29,39,30,27,31,38)(19,32,36,34,35,40,28,37);(9,12)(10,18)(11,16)(13,27)(14,25)(15,26)(17,21)(19,34)(20,33)(22,23)(24,30)(28,32)(29,38)(31,39)(35,37)(36,40);(1,3,4,8)(2,6,5,7)(9,13)(10,19)(11,24)(12,27)(14,31)(15,29)(16,30)(17,32)(18,34)(20,36)(21,28)(22,35)(23,37)(25,39)(26,38)(33,40)

Polycyclic presentation:

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

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, 279819);