#354631

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1632 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310050, D₄×#1765, #72310146
maximal normal subgroup of	#72310050, D₄×#1765, #72310146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261827);