#3348663

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1784 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)(5,6,10,8)(7,11,9,12)(13,14,18,15)(16,21,23,19)(17,20,26,22)(24,28,25,27)(29,30,34,38)(31,32,36,35)(33,41,42,37)(39,43,44,40);(5,7,10,9)(6,11,8,12)(14,15)(16,23)(20,22)(24,25)(31,36)(32,35)(39,44)(40,43);(5,8,10,6)(7,11,9,12)(13,15,18,14)(16,21,23,19)(17,20,26,22)(24,27,25,28)(29,31,34,36)(30,35,38,32)(33,39,42,44)(37,43,41,40);(1,2)(3,4)(5,9,10,7)(6,11,8,12)(13,16)(14,19)(15,21)(17,25)(18,23)(20,28)(22,27)(24,26)(29,32,34,35)(30,36,38,31)(33,40,42,43)(37,44,41,39);(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,28)(23,25)(29,33,34,42)(30,37,38,41)(31,39,36,44)(32,40,35,43)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=h,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3255859);