#392914

order	512
nilpotency class	3
isoclinism	isoclinic class of #32929
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287485, #72287546, #72287558
maximal normal subgroup of	#72287485, #72287546, #72287558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300110);