#361465

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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	C₂²×C₄⋊C₄
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1672 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)(9,10,15,19,14,17,13,24)(11,20,22,18,12,23,21,16);(2,4)(5,6)(7,8)(9,11,15,22,14,12,13,21)(10,16,24,23,17,18,19,20);(2,4)(6,8)(10,17)(16,18)(19,24)(20,23);(2,4)(9,12,15,21,14,11,13,22)(10,18,24,20,17,16,19,23)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1810, #72310129, #72310134, …
maximal normal subgroup of	D₄×#1810, #72310129, #72310134, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268661);