#556024

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1942 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

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, 463220);