#659273

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2774 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)(3,7,8,4)(9,10,14,15)(11,16,12,13)(17,18,21,22)(19,20,23,24)(25,26,29,31)(27,28,32,30);(2,6)(4,7)(9,11,14,12)(10,13,15,16)(18,22)(20,24)(25,27,29,32)(26,30,31,28);(1,3)(2,4)(5,8)(6,7)(9,10,14,15)(11,16,12,13)(17,19,21,23)(18,20,22,24)(25,28)(26,27)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,12,14,11)(10,13,15,16);(1,4)(2,3)(5,7)(6,8)(9,13)(10,11)(12,15)(14,16)(17,20)(18,23)(19,22)(21,24)(25,27)(26,28)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=fg,d^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 566469);