#5473768

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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×#2813	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2476 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)(24,28,33,32)(30,35,36,34);(7,11)(10,12)(13,14)(15,19)(16,17)(18,20)(22,26)(27,29)(28,32)(34,35);(6,9)(10,12)(14,17)(18,19)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36);(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)(24,30,33,36)(28,34,32,35);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=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, 5380964);