#6351043

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2877 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)(3,13)(4,9)(5,11)(6,10)(7,12)(8,14)(15,16)(17,18,23,28)(19,29,33,38)(20,34,35,26)(21,27,22,25)(24,32,40,31)(30,44,45,43)(36,41,37,42)(39,46,48,47);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(17,19,23,33)(18,24,28,40)(20,30,35,45)(21,32,22,31)(25,38,27,29)(26,39,34,48)(36,47,37,46)(41,44,42,43);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(18,25)(20,35)(21,22)(24,38)(26,42)(27,28)(29,40)(30,45)(31,32)(34,41)(39,43)(44,48);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(17,20,23,35)(18,26,28,34)(19,30,33,45)(21,36,22,37)(24,39,40,48)(25,41,27,42)(29,43,38,44)(31,46,32,47);(20,35)(26,34)(30,45)(36,37)(39,48)(41,42)(43,44)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258239);