#6443654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35168	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2453 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^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

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, 6350850);