#6462983

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370179);