#3440805

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

1607 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=fh,e^2=fg,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, 3348001);