#1281555

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	#14716	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266509, #72266510, #72266512, …
maximal normal subgroup of	#72266509, #72266510, #72266512, …
maximal quotient of	#72334843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1188751);