#372628

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

1106 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)(11,20)(12,24,28,17)(13,29)(14,18)(15,19)(16,22)(21,34,37,39)(23,32)(25,40,38,31)(26,35,27,36)(30,33);(2,5)(6,7)(9,11)(10,16)(12,25,28,38)(13,30)(14,22)(15,23)(17,34,24,39)(18,32)(19,33)(20,29)(21,27,37,26)(31,35,40,36);(9,12)(10,17)(11,21)(13,27)(14,26)(15,28)(16,31)(18,36)(19,24)(20,34)(22,38)(23,37)(25,30)(29,35)(32,39)(33,40);(1,3,4,8)(2,6,5,7)(9,11)(10,16)(12,21)(13,30)(14,22)(15,23)(17,31)(18,32)(19,33)(20,29)(24,40)(25,27)(26,38)(28,37)(34,35)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279824);