#6369189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1467 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)(23,29)(26,37)(28,45)(30,43)(31,48)(32,44)(33,41)(35,39)(36,42)(38,47)(40,46);(1,3,8,9)(2,4,13,5)(6,15,7,10)(11,16,12,14)(17,19,22,30)(18,23,25,39)(20,31,33,47)(21,35,37,29)(24,40,41,45)(26,43,34,27)(28,36,46,32)(38,44,48,42);(1,4,8,5)(2,9,13,3)(6,14,7,16)(10,11,15,12)(17,18)(19,23)(20,24)(21,26)(22,25)(27,29)(28,38)(30,39)(31,40)(32,42)(33,41)(34,37)(35,43)(36,44)(45,47)(46,48);(1,5,8,4)(2,10,13,15)(3,12,9,11)(6,14,7,16)(17,20,22,33)(18,24,25,41)(19,28,30,46)(21,36,37,32)(23,38,39,48)(26,44,34,42)(27,45,43,40)(29,47,35,31);(3,9)(4,5)(10,15)(14,16)(18,25)(23,39)(24,41)(26,34)(27,43)(38,48)(40,45)(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=ci,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283574, #72283588, #72283600, …
maximal normal subgroup of	#72283574, #72283588, #72283600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276385);