#1303184

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1724 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=fg,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=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, 1210380);