#6440190

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	#35195	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2185 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,g^c=gi

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, 6347386);