#6367238

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	#62200	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1578 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:

(1,2,5,8)(3,6,10,15)(4,7,12,9)(11,13,16,14)(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)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(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);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(19,28)(23,38)(27,43)(29,46)(30,35)(31,47)(39,48)(40,45);(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);(2,8)(3,10)(7,9)(11,16)(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,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	#72306994, #72307011, #72307019
maximal normal subgroup of	#72306994, #72307011, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274434);