#346479

order	512
nilpotency class	3
isoclinism	isoclinic class of #346446
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62826	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253675);