#6367293

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
prime radical	#3544	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1482 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,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	#72306999, #72307025
maximal normal subgroup of	#72306999, #72307025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274489);