#5025771

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
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

2622 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)(5,6,8,7)(9,10,13,11)(12,15,16,14)(17,18,22,23)(19,20,25,28)(21,29,32,24)(26,31,30,27);(9,10,13,11)(12,14,16,15)(18,23)(19,25)(24,29)(26,30);(6,7)(10,11)(14,15)(17,19)(18,20)(21,30)(22,25)(23,28)(24,31)(26,32)(27,29);(1,2)(3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,31)(22,28)(23,25)(24,30)(26,29)(27,32);(9,12,13,16)(10,14,11,15)(17,21)(18,24)(19,26)(20,27)(22,32)(23,29)(25,30)(28,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,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, 4932967);