#392316

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299512);