#6358088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358057
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	#61006	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1794 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265284);