#6365573

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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	#61914	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1492 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306859, #72306890
maximal normal subgroup of	#72306859, #72306890

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272769);