#557180

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

1726 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=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, 464376);