#347474

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#63045	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2235 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268465, #72271968
maximal normal subgroup of	#72268465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254670);