#4600724

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2066 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=fg,d^2=gh,e^2=f,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, 4507920);