#364016

order	512
nilpotency class	3
isoclinism	isoclinic class of #363904
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13957
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 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)(3,12,15,27)(4,16,18,8)(5,10)(6,11)(7,14,23,30)(9,17,25,31)(13,26)(19,20)(21,32)(22,29)(24,28);(1,3,5,14,6,15,19,30)(2,7,20,27,11,23,10,12)(4,13,17,28,18,29,31,32)(8,24,25,26,16,21,9,22);(1,4)(2,8)(3,13)(5,17)(6,18)(7,21)(9,10)(11,16)(12,26)(14,28)(15,29)(19,31)(20,25)(22,27)(23,24)(30,32);(1,4)(2,9,11,25)(3,13)(5,17)(6,18)(7,22,23,26)(8,10,16,20)(12,21,27,24)(14,28)(15,29)(19,31)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,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, 271212);