#3135433

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2805 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:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=fi,e^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3042629);