#6357591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357552
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	#3508	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1798 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264787);