#4416526

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2194 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)(7,11)(9,12)(15,19)(17,20)(21,22,25,28)(23,26,30,34)(24,31,32,27)(29,35,36,33);(5,6,8,10)(7,9,11,12)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,31)(25,28)(27,32)(29,35)(30,34)(33,36);(5,7)(6,9)(8,11)(10,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,30)(22,26,28,34)(24,29,32,36)(27,33,31,35);(5,7)(6,9)(8,11)(10,12)(21,24)(22,27)(23,29)(25,32)(26,33)(28,31)(30,36)(34,35)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=hi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef

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, 4323722);