#4606285

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1737 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)(7,11)(8,9)(10,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30)(31,32)(33,37)(34,35)(36,40)(38,39)(41,44)(42,43);(5,7,9,12)(6,10,8,11)(15,21)(19,22)(23,27)(26,28)(30,35)(31,36)(37,43)(38,41);(6,8)(10,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31,34,36)(30,32,35,40)(33,41,42,38)(37,44,43,39);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,36)(31,35)(33,39)(34,40)(37,41)(38,43)(42,44);(5,8,9,6)(7,11,12,10)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33,34,42)(30,37,35,43)(31,38,36,41)(32,39,40,44)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fgh,e^2=gh,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, 4513481);