#6372190

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3434 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279386);