#3531863

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(3,7)(5,8)(9,10,14,12)(11,15,13,16)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,38,35)(36,41,43,39)(37,40,46,42)(44,48,45,47);(9,11,14,13)(10,15,12,16)(18,19)(23,25)(24,26)(31,32)(34,35)(36,43)(40,42)(44,45);(1,2,4,6)(3,5,7,8)(9,12,14,10)(11,15,13,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,38,34)(36,41,43,39)(37,40,46,42)(44,47,45,48);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,15,12,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36)(34,39)(35,41)(37,45)(38,43)(40,48)(42,47)(44,46);(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=i,d^2=fgi,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, 3439059);