#2128567

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
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

2802 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)(5,6,7,8)(9,10,13,15)(11,12,16,14)(17,18,23,26)(19,24,22,32)(20,21,30,25)(27,31,29,28);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,19,23,22)(18,24,26,32)(20,27,30,29)(21,31,25,28);(1,2,3,4)(10,15)(12,14)(17,20)(18,25)(19,27)(21,26)(22,29)(23,30)(24,28)(31,32);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,25)(18,20,26,30)(19,28,22,31)(24,29,32,27);(1,2)(3,4)(9,12,13,14)(10,11,15,16)(17,22,23,19)(18,24,26,32)(20,29,30,27)(21,31,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=h,d^2=fg,e^2=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2035763);