#420654

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1075 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)(6,10)(11,12)(13,14,18,21,19,24,17,23)(15,20,26,27,22,16,25,28);(5,6,9,10)(7,11,8,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,27)(23,28);(2,4)(5,7,9,8)(6,11,10,12)(14,21)(15,25)(17,18)(22,26)(23,24)(27,28);(5,8,9,7)(6,11,10,12)(13,16)(14,22)(15,24)(17,28)(18,27)(19,20)(21,26)(23,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 327850);