#4035611

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
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

2678 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)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32);(5,6,7,8)(11,16)(13,15)(18,22)(20,29)(23,25)(26,30);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,32)(30,31);(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3942807);