#3825509

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

1673 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=fi,e^2=fi,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, 3732705);