#122163

order	512
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1446
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29359);