#3464939

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2651 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)(9,10)(11,16)(12,15)(13,14)(18,22)(19,25)(20,27)(30,32);(5,6,7,8)(9,11,14,12)(10,15,13,16)(19,25)(23,28)(26,31)(30,32);(5,6)(7,8)(10,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,2,3,4)(9,12,14,11)(10,15,13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32);(9,13,14,10)(11,16,12,15)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|b^2=fh,c^2=fg,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,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, 3372135);