#7561361

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2163 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)(7,11)(8,12)(9,10);(2,3)(6,10)(11,12)(13,14,18,22)(15,19,24,33)(16,26,27,36)(17,29,31,21)(20,30,35,28)(23,37,39,44)(25,41,40,34)(32,42,43,38);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,32)(21,34)(22,33)(26,37)(27,39)(28,38)(29,41)(30,42)(31,40)(35,43)(36,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16,18,27)(14,20,22,35)(15,23,24,39)(17,30,31,28)(19,32,33,43)(21,36,29,26)(25,42,40,38)(34,44,41,37);(6,10)(11,12)(15,24)(16,27)(19,33)(20,35)(25,40)(26,36)(28,30)(34,41)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7468557);