#347003

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2067 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254199);