#6444087

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×#3055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2298 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)(7,11)(8,12)(9,10)(14,18)(15,21)(19,23)(20,25)(22,27)(26,28);(1,2)(3,4)(6,10)(11,12)(13,14,17,18)(15,19,23,21)(16,20,24,25)(22,26,28,27);(2,4)(5,7,9,8)(6,11,10,12)(13,15,14,19,17,23,18,21)(16,22,20,26,24,28,25,27);(2,4)(5,8,9,7)(6,12,10,11)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27);(16,24)(20,25)(22,28)(26,27)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,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, 6351283);