#6443463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2036 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:

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

Polycyclic presentation:

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

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, 6350659);