#3750670

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1358 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=gh,d^2=i,e^2=hi,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, 3657866);