#6810814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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
prime radical	#42794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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)(6,10)(7,8)(13,14)(15,24)(16,28)(17,22)(18,23)(19,21)(20,27)(25,26);(5,6)(7,12)(8,11)(9,10)(13,14,18,23)(15,25,27,17)(16,21,19,28)(20,26,24,22);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,26,18,27,19,22)(14,20,28,17,23,24,21,25);(5,8,9,7)(6,11,10,12)(13,16,18,19)(14,21,23,28)(15,26,27,22)(17,20,25,24);(13,17,19,20,18,25,16,24)(14,22,21,27,23,26,28,15)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=fhi,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, 6718010);