#6365665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
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

1502 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)(3,7)(4,10)(5,14)(6,9)(8,11)(12,15)(13,16)(17,18,23,28)(19,29,33,40)(20,25,22,27)(21,26,36,44)(24,30,39,32)(31,45,47,42)(34,41,37,43)(35,48,46,38);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(17,19,23,33)(18,24,28,39)(20,32,22,30)(21,35,36,46)(25,40,27,29)(26,42,44,45)(31,37,47,34)(38,41,48,43);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,20,23,22)(18,25,28,27)(19,30,33,32)(21,34,36,37)(24,29,39,40)(26,41,44,43)(31,35,47,46)(38,45,48,42);(2,9)(7,15)(8,16)(10,14)(17,21)(18,26)(19,31)(20,34)(22,37)(23,36)(24,38)(25,41)(27,43)(28,44)(29,45)(30,35)(32,46)(33,47)(39,48)(40,42);(1,4,6,5)(2,10,9,14)(3,11,12,13)(7,8,15,16)(17,20,23,22)(18,27,28,25)(19,32,33,30)(21,34,36,37)(24,29,39,40)(26,43,44,41)(31,46,47,35)(38,45,48,42)

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306971, #72306975, #72307019
maximal normal subgroup of	#72306971, #72306975, #72307019

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272861);