#354709

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1112 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)(10,14,16,19)(11,17)(15,22,23,21)(18,24);(5,6,7,8)(9,10)(11,15)(12,19)(13,16)(14,20)(17,23)(18,22)(21,24);(1,2,3,4);(9,11,12,18,13,17,20,24)(10,15,19,22,16,23,14,21)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1773, #72310146, #72310152
maximal normal subgroup of	D₄×#1773, #72310146, #72310152

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261905);