#606119

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1876 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=ghi,d^2=fhi,e^2=fgi,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, 513315);