#1362179

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2794 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,d^2=g,e^2=fghi,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	#72304058, #72304093, #72304113
maximal normal subgroup of	#72304058, #72304093, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1269375);