#4933528

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2748 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)(5,6,9,10)(7,8,11,12)(15,19)(18,20)(21,22,26,24)(23,29,31,27)(25,28,34,32)(30,36,33,35);(5,7)(6,8)(9,11)(10,12)(14,17)(18,20)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(6,10)(7,11)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,35)(32,36)(33,34);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(21,24,26,22)(23,29,31,27)(25,32,34,28)(30,36,33,35);(5,8)(6,7)(9,12)(10,11)(13,15)(14,18)(16,19)(17,20)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,d^2=f,e^2=f,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, 4840724);