#2033598

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1940794);