#7518518

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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1995 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,27)(17,22)(18,23)(20,25)(21,29)(24,36)(26,35)(28,44)(30,41)(31,40)(32,37)(33,39)(34,42)(38,43);(1,2)(3,4)(6,9)(10,11)(13,15,17,25)(14,19,22,20)(16,28,29,43)(18,32,31,26)(21,38,27,44)(23,37,40,35)(24,33,42,30)(34,41,36,39);(2,4)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,36)(23,37)(27,34)(28,30)(29,42)(31,32)(33,43)(35,40)(38,41)(39,44);(2,4)(13,16,17,29)(14,21,22,27)(15,24,25,42)(18,33,31,30)(19,34,20,36)(23,41,40,39)(26,43,32,28)(35,44,37,38);(5,7,8,12)(6,10,9,11)(15,25)(19,20)(24,42)(26,32)(28,43)(34,36)(35,37)(38,44)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7425714);