#370540

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

1247 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)(11,21)(12,26,30,35)(13,19)(14,18)(15,20)(16,23)(17,28,36,29)(22,38,40,31)(24,33)(25,32)(27,37,39,34);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,27,30,39)(13,24)(14,23)(15,25)(17,34,36,37)(18,32)(19,21)(20,33)(22,29,40,28)(26,31,35,38);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,26)(20,36)(21,37)(23,27)(24,39)(25,40)(32,34)(33,38);(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,31)(18,32)(19,21)(20,33)(26,37)(27,29)(28,39)(30,40)(34,35)(36,38)

Polycyclic presentation:

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

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, 277736);