#6443673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=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, 6350869);