#359337

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6)(7,9)(8,10)(11,12)(13,14,18,19)(16,17,26,21)(22,27)(25,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,18,24)(14,20,19,23)(16,25,26,28)(17,22,21,27);(2,4)(6,10)(9,12)(13,16,14,21,18,26,19,17)(15,22,23,28,24,27,20,25);(13,17,19,26,18,21,14,16)(15,22,23,28,24,27,20,25)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266533);