#6671911

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2379 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:

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

Polycyclic presentation:

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

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, 6579107);