#6443882

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2225 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)(7,12)(8,10)(9,11)(13,14,18,21)(15,19,23,26)(16,25,20,17)(22,28,27,24);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,22)(17,24)(18,23)(20,27)(21,26)(25,28);(2,4)(13,16,17,14,18,20,25,21)(15,22,24,19,23,27,28,26);(2,4)(5,8,9,7)(6,10,11,12);(15,23)(19,26)(22,27)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351078);