#378966

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gi

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, 286162);