#4986235

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2106 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)(5,6,8,10)(7,9,12,11)(13,14,18,20)(15,19,17,16)(21,22)(23,29)(24,33)(25,34)(26,27)(28,32)(30,31)(35,36);(5,6)(7,11)(8,10)(9,12)(13,15,18,17)(14,19,20,16)(22,27)(25,33)(28,29)(31,35);(5,6,8,10)(7,9,12,11)(13,16,18,19)(14,17,20,15)(21,23,26,32)(22,28,27,29)(24,30,34,36)(25,31,33,35);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,25)(23,30)(26,34)(27,33)(28,31)(29,35)(32,36);(5,7,8,12)(6,9,10,11)(13,17,18,15)(14,16,20,19)(21,25)(22,24)(23,31)(26,33)(27,34)(28,30)(29,36)(32,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=gh,d^2=fi,e^2=fg,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, 4893431);