#6462991

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2292 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)(13,14,17,22,18,21,30,35)(15,23,25,32,26,36,29,19)(16,20,28,34,27,33,40,43)(24,37,39,42,38,44,41,31);(7,8)(10,12)(13,15,18,26)(14,19,21,32)(16,24,27,38)(17,29,30,25)(20,31,33,42)(22,36,35,23)(28,41,40,39)(34,44,43,37);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,24)(17,28)(18,27)(19,31)(21,33)(22,34)(23,37)(25,39)(26,38)(29,41)(30,40)(32,42)(35,43)(36,44);(5,8,9,7)(6,10,11,12)(16,27)(20,33)(24,38)(28,40)(31,42)(34,43)(37,44)(39,41);(14,21)(19,32)(20,33)(22,35)(23,36)(31,42)(34,43)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370187);