#222928

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=eh,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,d^c=di,g^c=gh

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, 130124);