#3153936

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3579 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232583, #72232628, #72232650
maximal normal subgroup of	#72232583, #72232628, #72232650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3061132);