#1033768

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2142 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=g,d^2=f,e^2=fh,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, 940964);