#2760921

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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

2188 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)(5,6,8,9)(7,11,12,10)(14,17)(18,19)(21,22)(23,28)(24,31)(25,27)(26,33)(29,34)(30,36)(32,35);(6,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,27,24)(22,28,25,31)(26,30,34,32)(29,35,33,36);(5,6,8,9)(7,10,12,11)(13,15,16,20)(14,18,17,19)(21,24,27,23)(22,28,25,31)(26,32,34,30)(29,35,33,36);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,19)(16,20)(17,18)(21,25,27,22)(23,28,24,31)(26,33,34,29)(30,35,32,36);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,26)(22,29)(23,30)(24,32)(25,33)(27,34)(28,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2668117);