#4639141

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fghi,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4546337);