#6809872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806527
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2199 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)(8,10)(9,12)(13,14,19,18)(15,20,24,27)(16,17,21,25)(22,23,26,28);(5,7)(6,9)(8,12)(10,11)(13,15,19,24)(14,20,18,27)(16,23,21,28)(17,26,25,22);(2,4)(7,12)(9,11)(13,16,18,25,19,21,14,17)(15,22,20,23,24,26,27,28);(6,10)(9,11)(13,14,19,18)(15,20,24,27)(16,17,21,25)(22,23,26,28);(13,17,14,21,19,25,18,16)(15,23,27,22,24,28,20,26)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717068);