#355543

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	C₂⁵	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1868 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)(15,18)(16,19,23,26)(17,20)(21,24,27,28)(22,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,21)(17,22)(19,24)(20,25)(23,27)(26,28);(6,9)(10,12)(13,16)(14,19,20,26)(15,21)(17,23)(18,24,25,28)(22,27);(5,7,8,11)(6,10,9,12)(15,22)(18,25)(21,27)(24,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,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, 262739);