#2761032

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2574 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)(7,11)(8,9)(10,12)(13,14,18,16)(15,17,20,19)(21,22)(23,29)(24,32)(25,28)(26,31)(27,34)(30,35)(33,36);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,29,31,26)(24,27,34,32)(30,36,33,35);(6,8)(10,11)(14,16)(15,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,36)(32,34)(33,35);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,18,14)(15,17,20,19)(21,23,25,31)(22,26,28,29)(24,33,34,30)(27,36,32,35);(5,7)(6,11)(8,10)(9,12)(13,17)(14,20)(15,16)(18,19)(21,24,25,34)(22,27,28,32)(23,30,31,33)(26,35,29,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2668228);