#6355958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355925
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62117	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1489 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,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, 6263154);