#355580

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 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)(7,9)(8,11)(10,12)(15,21,22,23)(18,25,26,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,20)(17,19)(21,25)(22,26)(23,28)(24,27);(6,7)(10,11)(13,15,16,23,17,22,24,21)(14,18,20,28,19,26,27,25);(5,8,9,12)(6,10,7,11)(14,19)(18,26)(20,27)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efg,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262776);