#6370439

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×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=g,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cfi,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, 6277635);