#358591

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1654 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=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, 265787);