#221392

order	512
nilpotency class	3
isoclinism	isoclinic class of #196038
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#120	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,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, 128588);