#359328

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
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×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 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:

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

Polycyclic presentation:

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

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, 266524);