#6378214

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#36684	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285410);