#560015

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1798 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fg,d^2=fh,e^2=g,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, 467211);