#6365571

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	#61913	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1496 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,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	#72306799, #72306801, #72306825
maximal normal subgroup of	#72306799, #72306801, #72306825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272767);