#6366960

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366870
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

1520 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)(3,11)(4,12)(5,8)(6,15)(7,10)(9,16)(13,14)(17,18,22,25)(19,27,30,37)(20,24,33,39)(21,34,35,26)(23,29,38,31)(28,41,43,47)(32,45,46,40)(36,42,48,44);(1,3,7,13)(2,5,10,4)(6,12,16,8)(9,14,15,11)(17,19,22,30)(18,23,25,38)(20,28,33,43)(21,29,35,31)(24,36,39,48)(26,37,34,27)(32,42,46,44)(40,47,45,41);(1,4)(2,3)(5,7)(6,14)(8,9)(10,13)(11,16)(12,15)(17,20,22,33)(18,24,25,39)(19,28,30,43)(21,32,35,46)(23,36,38,48)(26,40,34,45)(27,41,37,47)(29,42,31,44);(1,5)(2,8)(3,9)(4,7)(6,14)(10,12)(11,16)(13,15)(19,29)(21,35)(23,37)(26,34)(27,38)(28,42)(30,31)(32,46)(36,47)(40,45)(41,48)(43,44);(18,25)(19,30)(24,39)(26,34)(28,43)(29,31)(40,45)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306794, #72306795, #72306825, …
maximal normal subgroup of	#72306794, #72306795, #72306825, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274156);