#7564799

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	#45269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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)(5,6,9,10)(7,11,8,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,37)(25,32)(27,36)(28,33)(29,34)(30,35)(31,39)(38,42)(40,44)(41,43);(2,3)(6,10)(11,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,31)(21,23)(22,36)(27,30)(28,39)(29,38)(32,35)(33,37)(34,44)(40,41)(42,43);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,28)(18,29)(19,31)(21,33)(22,34)(23,37)(25,38)(26,39)(27,40)(30,41)(32,42)(35,43)(36,44);(1,3,4,2)(14,21)(15,25)(18,30)(19,32)(20,33)(23,36)(24,38)(26,27)(29,41)(31,42)(37,44)(39,40);(5,8,9,7)(6,11,10,12)(14,21)(15,26)(16,28)(22,35)(25,27)(29,41)(31,37)(42,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 7471995);