#444157

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

853 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)(8,12)(9,11)(13,14)(15,16)(17,21)(18,22)(19,26)(20,24)(23,25)(27,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,21)(15,24,26,28)(16,27,19,20)(18,23,25,22);(6,11)(10,12)(13,15,25,20,17,26,18,27)(14,19,23,28,21,16,22,24);(2,4)(5,8,9,7)(6,12,11,10)(13,16)(14,20)(15,22)(17,19)(18,28)(21,27)(23,26)(24,25)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^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,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, 351353);