#4952860

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1380 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fg,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4860056);