#6370406

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2338 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277602);