#805031

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 712227);