#5622656

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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	C₂×#2703	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2317 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,17)(18,20)(21,22,25,26)(23,29,31,27)(24,28,32,34)(30,36,33,35);(7,11)(10,12)(13,14)(15,19)(16,17)(18,20)(22,26)(24,32)(27,29)(30,33);(6,9)(10,12)(14,17)(18,19)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,31)(22,27,26,29)(24,33,32,30)(28,36,34,35);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,32)(22,28,26,34)(23,30,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=f,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5529852);