#7517284

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

1849 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)(13,14,18,21)(15,19,26,32)(16,20,27,35)(17,22,29,36)(23,30,38,43)(24,31,25,33)(28,34,40,44)(37,41,39,42);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,23)(17,24)(18,26)(20,30)(21,32)(22,31)(25,29)(27,38)(28,37)(33,36)(34,41)(35,43)(39,40)(42,44);(2,4)(13,16)(14,20)(15,23)(17,28)(18,27)(19,30)(21,35)(22,34)(24,37)(25,39)(26,38)(29,40)(31,41)(32,43)(33,42)(36,44);(2,4)(5,8,9,7)(6,10,11,12)(15,24)(17,29)(19,31)(22,36)(23,37)(25,26)(28,40)(30,41)(32,33)(34,44)(38,39)(42,43);(6,11)(10,12)(14,21)(16,27)(19,32)(22,36)(23,38)(28,40)(31,33)(37,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,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, 7424480);