#99783

order	512
nilpotency class	3
isoclinism	isoclinic class of #96924
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#121	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cgi,d^b=dh,f^b=fh

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, 6979);