#362278

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

888 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)(7,11)(8,10)(9,12)(13,14,19,15,18,22,17,21)(16,20,27,23,26,24,25,28);(5,7,9,8)(6,10,12,11)(13,15,19,22,18,21,17,14)(16,24,25,23,26,20,27,28);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,18,26)(14,20,22,24)(15,23,21,28)(17,25,19,27);(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,21,22,15)(16,25,26,27)(20,28,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269474);