#383611

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1562 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)(15,22)(16,20,26,19)(17,21)(18,25)(23,27,24,28);(1,2)(3,4)(5,6,8,9)(7,12,11,10)(13,15,17,25)(14,18,21,22)(16,23,26,24)(19,27,20,28);(2,4)(13,16)(14,19)(15,23)(17,26)(18,27)(20,21)(22,28)(24,25);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,24)(17,26)(18,27)(19,21)(22,28)(23,25)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,h^c=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290807);