#450211

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
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

1146 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)(5,6,9,8)(7,11,12,10)(13,14,20,24,18,22,19,23)(15,21,25,16,17,27,28,26);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,24)(17,18)(19,28)(20,25)(22,27)(23,26);(2,4)(5,8,9,6)(7,11,12,10)(13,16,18,26)(14,15,22,17)(19,21,20,27)(23,28,24,25);(2,4)(7,12)(10,11)(13,17)(14,16)(15,18)(19,28)(20,25)(21,24)(22,26)(23,27)

Polycyclic presentation:

2^9|a^2=efhi,c^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 357407);