#572198

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=i,e^2=hi,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, 479394);