#476257

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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	#23390	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1301 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383453);