#346580

order	512
nilpotency class	3
isoclinism	isoclinic class of #346550
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	#62902	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 253776);