#371589

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

929 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=e,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,h^c=hi

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, 278785);