#1030219

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2818 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=h,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, 937415);