#6370550

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×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2076 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=g,h^2=i,b^a=bf,d^a=df,e^a=ei,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, 6277746);