#4946181

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2128 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)(7,9)(8,10)(11,12)(13,14,17,19)(15,16,20,18)(22,26)(27,33)(28,31)(34,35);(13,15)(14,16)(17,20)(18,19)(21,22,25,26)(23,27,30,33)(24,31,32,28)(29,35,36,34);(5,6,8,10)(7,9,11,12)(13,15,17,20)(14,18,19,16)(21,22)(23,27)(24,28)(25,26)(29,34)(30,33)(31,32)(35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23,25,30)(22,27,26,33)(24,29,32,36)(28,34,31,35);(5,7,8,11)(6,9,10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,28)(23,29)(25,32)(26,31)(27,34)(30,36)(33,35)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=fg,d^2=f,e^2=gh,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, 4853377);