#385291

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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

973 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)(7,8)(9,11)(10,12)(13,14)(15,18)(16,22)(17,23)(19,21)(20,24)(25,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,20)(18,23)(19,21)(22,24)(25,28)(26,27);(5,8,9,10)(6,7,11,12)(13,15,18,14,19,23,17,21)(16,22,25,20,26,28,24,27);(6,11)(7,12)(13,16,17,24,19,26,18,25)(14,20,15,22,21,27,23,28)

Polycyclic presentation:

2^9|a^2=h,c^2=efi,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi

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, 292487);