#570390

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1706 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)(7,10,11,12)(14,18)(16,23)(19,24)(22,27)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(7,11)(10,12)(14,18)(15,21)(20,25)(22,27)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(1,2,4,3)(5,7)(6,10)(8,12)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,3,4,2)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,30)(31,36)(32,35)(33,37)(34,38)(39,44)(40,43)(41,42);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,44)(38,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 477586);