#4397636

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1914 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)(7,11)(9,12)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27)(29,30,33,36)(31,34,37,41)(32,39,40,35)(38,44,43,42);(14,15)(16,23)(20,22)(24,28)(31,37)(34,41)(38,43)(42,44);(1,2,3,4)(5,6,8,10)(7,9,11,12)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31,33,37)(30,34,36,41)(32,38,40,43)(35,42,39,44);(5,7,8,11)(6,9,10,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32,33,40)(30,35,36,39)(31,38,37,43)(34,42,41,44)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=fhi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4304832);