#1192824

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2586 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)(10,12)(11,15)(17,18,23,27)(19,25,20,24)(21,22,32,26)(28,29,30,31);(6,8)(17,19,23,20)(18,24,27,25)(21,28,32,30)(22,31,26,29);(5,6)(7,8)(9,10,13,12)(11,14,15,16)(17,20,23,19)(18,25,27,24)(21,28,32,30)(22,31,26,29);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(17,21)(18,26)(19,28)(20,30)(22,27)(23,32)(24,29)(25,31);(9,12,13,10)(11,16,15,14)(17,22,23,26)(18,21,27,32)(19,29,20,31)(24,30,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1100020);