#4598766

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 4194304
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2368 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)(7,11)(8,9)(10,12)(13,14,18,17)(15,16,20,19)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,36)(34,35);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,17)(18,20)(22,27)(23,28)(29,35)(30,33);(6,8)(10,11)(14,17)(15,20)(21,23,26,28)(22,24,27,32)(25,33,34,30)(29,36,35,31);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,28)(23,27)(25,31)(26,32)(29,33)(30,35)(34,36);(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,16,20,19)(21,25,26,34)(22,29,27,35)(23,30,28,33)(24,31,32,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=fg,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, 4505962);