#601060

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

1638 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)(5,6,8,9)(7,11,12,10)(14,18)(16,23)(19,24)(22,27)(30,31)(33,41)(35,37)(39,44);(6,9)(7,12)(15,21)(19,24)(22,27)(26,28)(29,30,34,31)(32,37,40,35)(33,38,41,36)(39,42,44,43);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,30)(32,35,40,37)(33,38,41,36)(39,42,44,43);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=ghi,d^2=fh,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, 508256);