#588324

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

1646 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)(5,6,9,8)(7,10,11,12)(14,18)(15,21)(16,23)(26,28)(29,30)(31,37)(32,40)(33,34)(35,39)(36,41)(38,42)(43,44);(1,3,4,2)(6,8)(10,12)(14,18)(15,21)(20,25)(22,27)(30,34)(32,41)(35,37)(38,44);(7,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,33,39)(30,35,34,37)(32,38,41,44)(36,42,40,43);(5,7)(6,10)(8,12)(9,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=gh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 495520);