#7517439

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

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,e^2=i,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,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, 7424635);