#6366772

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366710
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#18568	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1504 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283574
maximal normal subgroup of	#72283574

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273968);