#7517406

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1854 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)(7,12)(8,10)(9,11)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,37)(25,33)(27,32)(28,34)(29,35)(30,36)(31,39)(38,43)(40,42)(41,44);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,26)(18,25)(20,31)(21,23)(22,32)(27,30)(28,39)(29,40)(33,36)(34,37)(35,43)(38,41)(42,44);(2,4)(13,16)(14,20)(15,24)(17,28)(18,29)(19,31)(21,34)(22,35)(23,37)(25,40)(26,39)(27,38)(30,41)(32,43)(33,42)(36,44);(2,4)(5,8,9,7)(6,10,11,12)(15,25)(16,28)(18,30)(19,32)(20,34)(22,36)(23,33)(24,38)(26,27)(31,42)(37,43)(39,40);(6,11)(10,12)(14,21)(16,28)(19,23)(22,36)(24,39)(29,41)(32,33)(38,40)

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=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, 7424602);