#6444695

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2129 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,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^b=ei,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, 6351891);