#1790541

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2706 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)(3,6)(4,7)(5,8)(9,10,14,16)(11,13,12,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,28)(29,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18,20,22)(19,21,23,24)(25,27,29,32)(26,30,31,28);(3,8)(5,6)(10,16)(11,12)(19,23)(21,24)(26,31)(28,30);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,13,16,15)(17,19,20,23)(18,21,22,24)(25,28,29,30)(26,27,31,32);(1,2)(3,6)(4,7)(5,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232650, #72232691, #72232696, …
maximal normal subgroup of	#72232650, #72232691, #72232696, …
maximal quotient of	#72330768

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1697737);