#7520490

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#45316	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 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)(7,11)(8,10)(9,12)(13,14)(15,27)(16,32)(17,36)(18,24)(19,26)(20,25)(21,29)(22,34)(23,33)(28,44)(30,40)(31,41)(35,43)(37,42)(38,39);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,33)(17,34)(18,29)(19,31)(20,30)(22,42)(23,44)(24,27)(25,41)(26,40)(28,38)(32,39)(35,37)(36,43);(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,17)(18,28)(19,35)(20,34)(21,39)(23,27)(24,43)(25,44)(26,32)(29,37)(30,38)(31,33)(36,40)(41,42);(2,4)(13,17,18,37)(14,23,24,39)(15,28,29,16)(19,38,20,33)(21,22,27,43)(25,42,26,36)(30,34,31,35)(32,40,44,41);(5,8,9,7)(6,11,12,10)(14,24)(15,29)(16,28)(23,39)(25,26)(30,31)(34,35)(36,42)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,d^2=i,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=cg,d^b=dh,e^b=ei,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427686);