#7520841

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#25614	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=i,e^2=f,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=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7428037);