#1029915

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2920 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

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, 937111);