#6459662

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3283 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)(7,9)(8,10)(11,12)(13,14,19,22)(15,17,21,18)(16,24,27,20)(23,28,25,26);(5,7)(6,9)(8,11)(10,12)(13,15,18,14,19,21,17,22)(16,25,28,24,27,23,26,20);(1,2,3,4)(6,10)(7,11)(13,16)(14,20)(15,23)(17,26)(18,28)(19,27)(21,25)(22,24);(7,11)(9,12)(13,17,19,18)(14,21,22,15)(16,26,27,28)(20,25,24,23);(16,27)(20,24)(23,25)(26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366858);