#1013961

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2096 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=g,d^2=f,e^2=f,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, 921157);