#1085237

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=fg,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, 992433);