#2120754

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2264 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=fgh,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 2027950);