#406459

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1458 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,27)(24,25)(26,28);(6,9)(10,12)(13,14,18,22,17,21,16,20)(15,19,27,23,26,24,25,28);(1,2,3,4)(13,14,17,21)(15,24,26,19)(16,20,18,22)(23,27,28,25);(5,7,8,11)(6,10,9,12)(13,15,16,25,17,26,18,27)(14,19,20,28,21,24,22,23)

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226919, #72226928, #72226937
maximal normal subgroup of	#72226919, #72226928, #72226937

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313655);