#1392149

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1954 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)(6,7)(10,14)(11,12)(19,23)(22,24)(27,29)(28,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(25,26,30,32)(27,31,29,28);(9,11,13,12)(10,15,14,16)(17,18,21,20)(19,22,23,24)(25,27,30,29)(26,31,32,28);(1,2)(3,7)(4,5)(6,8)(9,12,13,11)(10,15,14,16)(17,19)(18,22)(20,24)(21,23)(25,28)(26,27)(29,32)(30,31);(1,3,4,8)(2,6,5,7)(9,10)(11,15)(12,16)(13,14)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,28,32,31)

Polycyclic presentation:

2^9|b^2=gh,c^2=fhi,d^2=h,e^2=hi,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, 1299345);