#396600

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

1167 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303796);