#6357723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3499	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1803 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,e^c=eh,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264919);