#354529

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,f^c=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261725);