#7518131

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1827 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)(7,12)(8,10)(9,11)(13,14,18,20)(15,25,29,37)(16,30,33,39)(17,24,19,23)(21,28,35,26)(22,34,36,31)(27,38,32,40)(41,42,44,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,20)(15,26)(16,31)(17,19)(21,37)(22,39)(25,35)(28,29)(30,36)(33,34)(38,40)(42,43);(13,15,18,29)(14,21,20,35)(16,32,33,27)(17,26,19,28)(22,40,36,38)(23,25,24,37)(30,44,39,41)(31,43,34,42);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,27)(17,31)(18,33)(19,34)(20,36)(21,38)(23,30)(24,39)(25,41)(26,42)(28,43)(29,32)(35,40)(37,44);(6,11)(10,12)(13,17,18,19)(14,23,20,24)(15,28,29,26)(16,31,33,34)(21,37,35,25)(22,30,36,39)(27,43,32,42)(38,44,40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,e^2=i,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,e^b=ei,g^b=gi,h^b=hi,d^c=di,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, 7425327);