#6817252

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fghi,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724448);