#5213650

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
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	C₂×#2812	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5120846);