#6466180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2488 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6373376);