#6367490

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1520 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306848, #72306853, #72306887
maximal normal subgroup of	#72306848, #72306853, #72306887

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274686);