#354723

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1664 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=fg,b^a=be,c^a=cf,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261919);