#7514838

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422034);