#600117

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

1630 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)(7,11)(10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30,33,36)(31,37,39,34)(32,40,42,35)(38,43,41,44);(6,9)(10,12)(14,15)(17,25)(19,21)(23,28)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(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)(30,34)(32,38)(33,39)(35,43)(36,37)(40,44)(41,42);(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,23,25,28)(20,26,22,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(1,3,4,2)(13,17)(14,20)(15,22)(16,23)(18,25)(19,26)(21,27)(24,28)(29,32)(30,35)(31,38)(33,42)(34,43)(36,40)(37,44)(39,41)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fghi,d^2=fg,e^2=fi,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, 507313);