#6378191

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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

1834 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)(13,14,17,22,18,20,29,33)(15,19,25,32,23,30,37,41)(16,26,27,40,28,21,34,35)(24,38,39,44,36,31,42,43);(5,6)(7,10)(8,9)(11,12)(15,23)(19,30)(24,36)(25,37)(31,38)(32,41)(39,42)(43,44);(2,4)(6,9)(10,12)(13,15)(14,19)(16,24)(17,25)(18,23)(20,30)(21,31)(22,32)(26,38)(27,39)(28,36)(29,37)(33,41)(34,42)(35,43)(40,44);(5,7,8,11)(6,10,9,12)(14,20)(15,23)(21,26)(22,33)(24,36)(25,37)(35,40)(39,42);(13,16,17,27,18,28,29,34)(14,21,22,35,20,26,33,40)(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=dfi,e^a=ei,c^b=cfi,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, 6285387);