#365935

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,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, 273131);