#355547

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fi,g^c=gi

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, 262743);