#1241769

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2532 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=fg,e^2=fghi,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, 1148965);