#7560374

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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1850 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7467570);