#4395344

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
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

1761 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)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,35)(31,36,38,33)(32,40,41,37)(39,42,44,43);(5,6,8,10)(7,9,11,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,38)(32,41)(42,43);(1,2,3,4)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31)(30,36)(32,39)(33,35)(34,38)(37,43)(40,42)(41,44);(5,7,8,11)(6,9,10,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,41)(30,37,35,40)(31,39,38,44)(33,42,36,43);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33,34,36)(30,38,35,31)(32,42,41,43)(37,44,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,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, 4302540);