#6438211

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345407);