#6444074

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2240 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)(15,21)(16,22)(18,24)(20,25)(23,27)(26,28);(1,2)(3,4)(6,10)(11,12)(13,14)(15,18)(16,20)(17,19)(21,24)(22,25)(23,26)(27,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,22,27,17,23,16,21)(14,18,25,28,19,26,20,24);(2,4)(5,8,9,7)(6,12,10,11)(15,23)(18,26)(21,27)(24,28);(14,19)(18,26)(20,25)(24,28)

Polycyclic presentation:

2^9|c^2=fgi,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, 6351270);