#6368940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368846
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	#18713	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1465 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)(4,7,13,11)(5,15,16,9)(17,18)(19,27)(20,24)(21,34)(22,25)(23,30)(26,37)(28,45)(29,43)(31,48)(32,44)(33,41)(35,39)(36,42)(38,47)(40,46);(1,3,6,12)(2,4,8,13)(5,7,16,11)(9,14,15,10)(17,19,22,29)(18,23,25,39)(20,31,33,47)(21,35,37,30)(24,40,41,45)(26,43,34,27)(28,36,46,32)(38,44,48,42);(1,4,6,13)(2,3,8,12)(5,10,16,14)(7,9,11,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,38)(29,39)(31,40)(32,42)(33,41)(34,37)(35,43)(36,44)(45,47)(46,48);(1,4,6,13)(2,7,8,11)(3,9,12,15)(5,14,16,10)(17,20,22,33)(18,24,25,41)(19,28,29,46)(21,36,37,32)(23,38,39,48)(26,44,34,42)(27,45,43,40)(30,47,35,31);(2,8)(4,13)(9,15)(10,14)(18,25)(19,29)(24,41)(26,34)(28,46)(30,35)(31,47)(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=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,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	#72306601, #72306631, #72306707
maximal normal subgroup of	#72306601, #72306631, #72306707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276136);