#4606928

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1898 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgh,c^2=g,e^2=f,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, 4514124);