#6466489

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi

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, 6373685);