#346477

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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,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, 253673);