#2046699

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1592 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)(3,8)(4,5)(6,7)(10,12)(11,15)(17,18)(19,23)(20,24)(21,22);(2,7)(5,8)(11,15)(14,16)(18,21)(19,24)(25,26,29,32)(27,30,28,31);(1,3,6,4)(2,8,7,5)(9,10,13,12)(11,14,15,16)(17,19,22,24)(18,23,21,20)(25,26)(27,30)(28,31)(29,32);(1,4,6,3)(2,5,7,8)(9,11,13,15)(10,14,12,16)(17,20)(18,19)(21,24)(22,23)(25,27,29,28)(26,30,32,31);(1,5)(2,3)(4,7)(6,8)(9,12,13,10)(11,16,15,14)(17,21,22,18)(19,20,24,23)(25,28,29,27)(26,31,32,30)

Polycyclic presentation:

2^9|b^2=ghi,c^2=fg,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1953895);