#348484

order	512
nilpotency class	3
isoclinism	isoclinic class of #348478
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29810	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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:

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

Polycyclic presentation:

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

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, 255680);