#362534

order	512
nilpotency class	3
isoclinism	isoclinic class of #362481
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2293
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13917
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

869 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)(3,15,5,23)(4,10,22,29)(6,12)(7,14)(8,13)(9,17,11,18)(16,26)(19,27)(20,32)(21,28,24,30)(25,31);(1,3,7,5)(2,9)(4,19)(6,17,8,18)(10,26,29,32)(11,14)(12,15)(13,23)(16,24)(20,21)(22,31)(25,28,27,30);(1,4)(2,10)(3,16)(5,20)(6,21)(7,22)(8,24)(9,25)(11,27)(12,28)(13,30)(14,29)(15,26)(17,19)(18,31)(23,32);(1,5,7,3)(2,11)(4,20,22,16)(6,18,8,17)(9,14)(10,27)(12,23)(13,15)(19,21,31,24)(25,29)(26,30)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,g^c=gi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269730);