#346491

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

2158 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)(17,18)(19,31)(20,36)(21,27)(22,29)(23,28)(24,30)(25,34)(26,37)(32,40)(33,47)(35,44)(38,43)(39,42)(41,46)(45,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,25)(20,32)(21,33)(22,39)(23,34)(24,35)(26,40)(27,41)(28,44)(29,31)(30,42)(36,48)(37,46)(38,45)(43,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,24,21)(18,26,30,43)(19,32,35,33)(22,37,23,38)(25,40,42,47)(27,28,36,29)(31,41,44,48)(34,45,39,46);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,20)(18,27,30,36)(19,33,35,32)(22,37,23,38)(25,41,42,48)(26,28,43,29)(31,40,44,47)(34,45,39,46)

Polycyclic presentation:

2^9|c^2=h,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, 253687);