#4606284

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

1752 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)(5,6,9,7)(8,10,12,11)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(30,34)(31,37)(36,40)(39,43);(6,7)(10,11)(15,21)(19,22)(23,27)(26,28)(29,30,33,34)(31,38,37,35)(32,36,41,40)(39,44,43,42);(5,7,9,6)(8,10,12,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,38)(39,44)(42,43);(1,2)(3,4)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(5,8,9,12)(6,10,7,11)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,32)(30,36)(31,39)(33,41)(34,40)(35,42)(37,43)(38,44)

Polycyclic presentation:

2^9|a^2=gh,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, 4513480);