#400863

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

857 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)(7,11)(8,12)(9,10)(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)(6,10)(11,12)(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)(5,7,9,8)(6,11,10,12)(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);(5,8,9,7)(6,12,10,11)(13,17,19,34)(14,22,24,40)(15,27,28,43)(16,29,30,42)(18,32,26,33)(20,36,37,44)(21,38,25,41)(23,39,35,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=e,d^2=ef,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, 308059);