#6368785

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368766
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306659, #72306694, #72306703
maximal normal subgroup of	#72306659, #72306694, #72306703

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275981);