#1505935

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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2762 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=gi,d^2=fgh,e^2=gi,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, 1413131);