#1302670

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2134 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=fgh,d^2=fgi,e^2=fgh,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, 1209866);