#6811554

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2147 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,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^c=ei,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, 6718750);