#382450

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2668 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)(8,11)(10,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25);(1,2)(3,4)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27);(2,4)(6,9,11,10)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28);(16,24)(19,26)(22,28)(25,27)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289646);