#6369202

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
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

1544 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,c^b=ci,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

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306675, #72306681, #72306684, …
maximal normal subgroup of	#72306675, #72306681, #72306684, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276398);