#6440192

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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2192 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)(13,14,17,22)(15,23,24,19)(16,21,18,27)(20,26,28,25);(1,2)(3,4)(6,9)(10,12)(13,15,17,24)(14,19,22,23)(16,26,18,25)(20,27,28,21);(2,4)(13,15)(14,20)(16,25)(17,24)(18,26)(19,21)(22,28)(23,27);(2,4)(15,24)(19,23)(20,28)(25,26);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,27)(15,25,24,26)(19,28,23,20)

Polycyclic presentation:

2^9|a^2=i,b^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, 6347388);