#378210

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 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)(7,11)(8,12)(9,10)(14,20)(15,23,26,27)(16,22)(21,24,28,25);(1,2)(3,4)(6,10)(11,12)(13,14,18,22,17,20,19,16)(15,24,23,21,26,25,27,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,17,26)(14,21,20,28)(16,25,22,24)(18,27,19,23);(5,8,9,7)(6,11,10,12)(13,16,19,20,17,22,18,14)(15,25,23,28,26,24,27,21)

Polycyclic presentation:

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

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, 285406);