#431479

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1131 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)(7,11)(8,10)(9,12)(13,14,18,24,20,26,19,25)(15,27,28,16,22,23,17,21);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,26)(17,18)(19,28)(20,22)(23,24)(25,27);(2,4)(6,10)(9,11)(13,16,18,23,20,21,19,27)(14,22,24,17,26,15,25,28);(7,12)(9,11)(13,17,19,22,20,28,18,15)(14,23,25,16,26,27,24,21)

Polycyclic presentation:

2^9|a^2=ef,c^2=ef,d^2=efi,e^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=de,e^b=ei,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, 338675);