#1101039

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2462 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=g,e^2=f,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, 1008235);