#6817496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3154	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fgh,d^2=f,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724692);