#4416581

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
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

2050 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)(7,10,12,11)(15,19)(17,20)(24,31)(27,34)(29,35)(32,36);(6,9)(10,11)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,32,35,36);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,18)(15,17,19,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(13,15)(14,17)(16,19)(18,20)(21,23,25,30)(22,26,28,33)(24,29,31,35)(27,32,34,36);(5,7,8,12)(6,10,9,11)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=i,d^2=fh,e^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, 4323777);