#354953

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1572 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)(6,8)(9,10)(11,21)(12,13)(14,18)(15,20)(16,17)(19,24)(22,23);(5,6,7,8)(9,11,15,22,14,17,24,13)(10,16,20,12,18,21,19,23);(9,12)(10,17)(11,18)(13,20)(14,23)(15,21)(16,24)(19,22);(9,13,24,17,14,22,15,11)(10,12,19,16,18,23,20,21)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310083, D₄×#1765, #72310146
maximal normal subgroup of	#72310083, D₄×#1765, #72310146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262149);