#7520772

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1829 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,e^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=cgi,d^b=dh,g^b=gi,h^b=hi,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, 7427968);