#581476

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgh,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^c=ef

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, 488672);