#6352396

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352341
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62879	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2637 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259592);