#6381077

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=df,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6288273);