#7517982

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	C₂×#3307	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,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, 7425178);