#7517287

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

1845 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)(13,14,18,23)(15,25,29,20)(16,21,31,38)(17,22,19,24)(26,41,43,34)(27,42,44,35)(28,40,33,36)(30,37,32,39);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,29)(14,20,23,25)(16,26,31,43)(17,28,19,33)(21,34,38,41)(22,36,24,40)(27,32,44,30)(35,39,42,37);(2,4)(13,16)(14,21)(15,26)(17,32)(18,31)(19,30)(20,34)(22,39)(23,38)(24,37)(25,41)(27,33)(28,44)(29,43)(35,40)(36,42);(2,4)(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,27)(17,30)(18,31)(19,32)(20,35)(22,37)(23,38)(24,39)(25,42)(26,33)(28,43)(29,44)(34,40)(36,41);(6,11)(10,12)(13,17,18,19)(14,22,23,24)(15,28,29,33)(16,30,31,32)(20,36,25,40)(21,37,38,39)(26,27,43,44)(34,35,41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,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, 7424483);