#547119

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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)(13,14,17,18)(15,19,21,22)(16,24,25,20)(23,28,27,26)(29,30,33,36)(31,34,39,37)(32,35,40,42)(38,41,44,43);(6,9)(10,12)(14,18)(15,21)(20,24)(23,27)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,6,8,9)(7,10,11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,41)(40,42)(43,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27)(29,31)(30,34)(32,38)(33,39)(35,41)(36,37)(40,44)(42,43);(13,16,17,25)(14,20,18,24)(15,23,21,27)(19,26,22,28)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gh,d^2=hi,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 454315);