#96753

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=ch,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3949);