#550573

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

1702 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)(13,14,18,15)(16,19,24,21)(17,22,25,20)(23,27,28,26)(29,30)(31,37)(32,40)(33,34)(35,38)(36,41)(39,42)(43,44);(1,2,4,3)(6,9)(10,12)(14,15)(19,21)(20,22)(26,27)(30,34)(31,38)(36,40)(39,44);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,26,28,27)(29,31)(30,35)(32,39)(33,38)(34,37)(36,42)(40,43)(41,44);(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,37)(32,36)(33,34)(35,38)(39,43)(40,41)(42,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,40)(31,39,38,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fhi,d^2=fg,e^2=hi,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, 457769);