#432184

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,12)(7,10,8,11)(13,14,17,21)(15,19,24,34)(16,26,28,20)(18,31,30,22)(23,39,40,33)(25,42,41,35)(27,36,43,44)(29,37,32,38);(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,27,28,43)(17,24)(18,25)(20,36,26,44)(21,34)(22,35)(23,29,40,32)(30,41)(31,42)(33,37,39,38);(5,7,9,8)(6,11,12,10)(13,16,17,28)(14,20,21,26)(15,23,24,40)(18,32,30,29)(19,33,34,39)(22,38,31,37)(25,43,41,27)(35,44,42,36);(2,4)(5,8,9,7)(6,11,12,10)(15,24)(18,30)(19,34)(22,31)(23,40)(29,32)(33,39)(37,38)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=f,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 339380);