#354700

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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

1206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226894, #72226903, #72226904
maximal normal subgroup of	#72226894, #72226903, #72226904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261896);