#6450072

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3038 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=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, 6357268);