#7526037

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(19,27)(20,31)(23,28)(24,40)(30,43)(32,42)(34,37)(39,44);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,25)(19,28,27,23)(20,32,31,42)(22,36,35,26)(24,37,40,34)(29,45,33,48)(30,39,43,44)(38,46,41,47);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,21,27)(18,23,25,28)(20,33,31,29)(22,30,35,43)(24,41,40,38)(26,39,36,44)(32,45,42,48)(34,47,37,46);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,21,31)(18,24,25,40)(19,29,27,33)(22,37,35,34)(23,38,28,41)(26,42,36,32)(30,47,43,46)(39,45,44,48);(2,6)(7,8)(19,27)(20,31)(23,28)(24,40)(30,43)(32,42)(34,37)(39,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7433233);