#569253

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1698 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)(13,14,18,19)(15,20,21,16)(17,22,25,26)(23,27,28,24)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(6,9)(10,12)(14,19)(15,21)(17,25)(24,27)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,23)(18,21)(22,27)(24,26)(25,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(1,3,4,2)(13,16,18,20)(14,21,19,15)(17,24,25,27)(22,28,26,23)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 476449);