#1452049

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2273 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)(3,7)(4,8)(5,6)(9,10,14,13)(11,12,16,15)(18,20)(19,24)(25,26,28,30)(27,31,32,29);(2,6)(4,7)(9,11,14,16)(10,15,13,12)(17,18)(19,23)(20,21)(22,24);(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,23)(20,22)(21,24)(25,27,28,32)(26,29,30,31);(1,4,5,7)(2,8,6,3)(9,13,14,10)(11,15,16,12)(17,20,21,18)(19,23,24,22)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1359245);