#6364061

order	512
nilpotency class	3
isoclinism	isoclinic class of #6363930
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	#61910	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1522 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)(3,14)(4,10)(5,13)(6,12)(7,11)(8,15)(9,16)(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);(1,3,7,9)(2,8,11,4)(5,10,6,15)(12,16,13,14)(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);(1,4)(2,9)(3,11)(5,16)(6,14)(7,8)(10,12)(13,15)(19,28)(23,38)(27,43)(29,46)(30,35)(31,47)(39,48)(40,45);(1,4)(2,10)(3,13)(5,14)(6,16)(7,8)(9,12)(11,15)(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);(2,11)(3,9)(10,15)(12,13)(18,25)(23,38)(24,41)(26,34)(27,43)(39,48)(40,45)(42,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,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

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306808, #72306814, #72306825
maximal normal subgroup of	#72306808, #72306814, #72306825

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6271257);