#2309557

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2643 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280828, #72280933, #72281381
maximal normal subgroup of	#72280828, #72280933, #72281381

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2216753);