#4606256

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1864 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)(6,9)(7,11)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(30,31)(32,39)(33,41)(42,43);(5,6)(7,10)(8,9)(11,12)(15,21)(19,22)(23,27)(26,28)(29,30,34,31)(32,37,39,35)(33,36,41,38)(40,43,44,42);(7,11)(10,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4513452);