#6368752

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368694
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

1564 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)(3,9)(4,6)(5,14)(7,11)(8,15)(10,13)(12,16)(17,18)(19,30)(20,26)(21,29)(22,28)(23,27)(24,34)(25,37)(31,40)(32,47)(33,46)(35,42)(36,45)(38,41)(39,43)(44,48);(1,3)(2,7)(4,13)(5,10)(6,12)(8,16)(9,14)(11,15)(17,19,23,33)(18,24,27,41)(20,35,36,31)(21,38,22,34)(25,42,43,40)(26,44,45,47)(28,46,29,30)(32,39,48,37);(1,4,6,2)(3,7,12,13)(5,14,15,8)(9,10,16,11)(17,20)(18,25)(19,31)(21,37)(22,39)(23,36)(24,40)(26,28)(27,43)(29,45)(30,44)(32,34)(33,35)(38,48)(41,42)(46,47);(3,10)(5,15)(7,16)(8,14)(9,13)(11,12)(17,20)(18,26)(19,32)(21,39)(22,37)(23,36)(24,40)(25,28)(27,45)(29,43)(30,47)(31,34)(33,48)(35,38)(41,42)(44,46);(2,4)(7,13)(8,14)(9,16)(18,27)(19,33)(20,36)(28,29)(34,38)(37,39)(40,42)(44,47)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ch,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	#72306650, #72306668, #72306697
maximal normal subgroup of	#72306650, #72306668, #72306697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6275948);