#368547

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=eg,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275743);