#6463825

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1954 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,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, 6371021);