#1287179

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	#14762	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=ghi,e^2=fg,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	#72304069, #72304093, #72304111
maximal normal subgroup of	#72304069, #72304093, #72304111

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1194375);