#6807150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2068 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:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714346);