#386189

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1188 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)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,20,24,18,22,19,23)(15,17,26,28,25,21,16,27);(5,7)(6,8)(9,12)(10,11)(13,15)(14,21)(16,19)(17,22)(18,25)(20,26)(23,28)(24,27);(2,4)(5,8,10,9)(6,11,12,7)(13,16,20,25,18,26,19,15)(14,17,24,27,22,21,23,28);(2,4)(5,9,10,8)(6,11,12,7)(13,17)(14,15)(16,24)(18,21)(19,28)(20,27)(22,25)(23,26)

Polycyclic presentation:

2^9|a^2=h,c^2=eg,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,h^c=hi,f^d=fi,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, 293385);