#6817004

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2244 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)(5,6,9,11)(7,12,8,10)(13,14)(15,24)(16,23)(17,21)(18,22)(19,26)(20,27)(25,28);(5,6)(7,10)(8,12)(9,11)(13,15,18,26)(14,19,22,24)(16,25,27,17)(20,28,23,21);(1,2,3,4)(13,15,16,17,18,26,27,25)(14,19,23,28,22,24,20,21);(5,7,9,8)(6,10,11,12)(13,16,18,27)(14,20,22,23)(15,25,26,17)(19,28,24,21);(5,8,9,7)(6,10,11,12)(13,17,27,15,18,25,16,26)(14,21,20,24,22,28,23,19)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=ghi,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,d^c=di,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, 6724200);