#1304761

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3444 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)(3,4)(5,7)(6,8)(10,14)(11,15)(18,21)(19,23)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,12,15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,32)(30,31);(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,6)(2,3,7,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,29)(28,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31)

Polycyclic presentation:

2^9|b^2=gi,d^2=fhi,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	#72304116
maximal normal subgroup of	#72304116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1211957);