#383392

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

1649 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=i,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, 290588);