#1648412

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1726 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1555608);