#553973

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

1718 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)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(30,34)(31,37)(36,40)(39,43);(6,9)(10,12)(14,15)(17,25)(19,21)(23,28)(29,30,33,34)(31,38,37,35)(32,40,41,36)(39,42,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,27,28,26)(29,30)(31,35)(32,36)(33,34)(37,38)(39,42)(40,41)(43,44);(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(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)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,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, 461169);