#6685646

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2057 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)(7,10)(8,9)(11,12)(13,14,19,20)(15,23,26,21)(16,18,22,17)(24,25,27,28);(6,9)(7,11)(13,15)(14,21)(16,25)(17,27)(18,24)(19,26)(20,23)(22,28);(1,2,3,4)(13,16)(14,18)(15,24)(17,20)(19,22)(21,28)(23,25)(26,27);(6,9)(10,12)(13,17,19,18)(14,22,20,16)(15,24,26,27)(21,25,23,28);(5,7,8,11)(6,10,9,12)(13,18,19,17)(14,22,20,16)(15,25,26,28)(21,24,23,27)

Polycyclic presentation:

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

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, 6592842);