#7522580

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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2154 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,31)(17,35)(18,24)(19,25)(20,26)(22,33)(23,32)(27,42)(28,39)(29,40)(30,41)(34,38)(36,44)(37,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,32)(17,33)(18,28)(19,29)(20,30)(22,35)(23,31)(24,39)(25,40)(26,41)(27,36)(34,37)(38,43)(42,44);(13,16,20,27)(14,22,26,38)(15,17,30,37)(18,34,19,33)(21,23,41,44)(24,31,25,42)(28,32,29,36)(35,39,43,40);(2,4)(5,7,9,8)(6,11,12,10)(13,17)(14,23)(15,27)(16,30)(18,36)(19,32)(20,37)(21,38)(22,41)(24,35)(25,43)(26,44)(28,34)(29,33)(31,39)(40,42);(5,8,9,7)(6,11,12,10)(13,18,20,19)(14,24,26,25)(15,28,30,29)(16,33,27,34)(17,36,37,32)(21,39,41,40)(22,42,38,31)(23,35,44,43)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=f,e^2=fi,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,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429776);