#4613818

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,d^2=fh,e^2=fh,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, 4521014);