#6454256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2266 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi

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, 6361452);