#6349960

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349901
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	C₄²	C₂⁵
prime radical	C₂×#3547	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2757 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:

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

Polycyclic presentation:

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

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, 6257156);