#7521853

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	#25797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=fi,e^2=f,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, 7429049);