#370554

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

1238 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)(4,8)(5,7)(9,10,14,20,15,21,23,19)(11,22,26,36,27,40,25,16)(12,28,30,37,24,35,31,17)(13,32,29,39,34,38,33,18);(1,3,5,4)(2,6,7,8)(9,11,23,25,15,27,14,26)(10,16,19,40,21,36,20,22)(12,29,31,13,24,33,30,34)(17,18,35,38,37,39,28,32);(1,4,5,3)(2,6,7,8)(9,12,15,24)(10,17,21,37)(11,13,27,34)(14,30,23,31)(16,32,36,38)(18,22,39,40)(19,35,20,28)(25,33,26,29);(9,13)(10,18)(11,24)(12,27)(14,29)(15,34)(16,35)(17,22)(19,38)(20,32)(21,39)(23,33)(25,30)(26,31)(28,36)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^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, 277750);