#426497

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 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)(5,6,9,11)(7,10,8,12)(13,14,17,21)(15,24,16,19)(18,22,28,23)(20,27,25,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,21)(15,25,16,20)(18,23,28,22)(19,27,24,26);(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,17)(18,27)(20,22)(21,24)(23,25)(26,28);(2,4)(13,16)(14,20)(15,17)(18,27)(19,22)(21,25)(23,24)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333693);