#346490

order	512
nilpotency class	3
isoclinism	isoclinic class of #346482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234939
maximal normal subgroup of	#72234939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253686);