#6368789

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

1493 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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, #72306699, #72306707
maximal normal subgroup of	#72306659, #72306699, #72306707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275985);