#382132

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14319
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(3,13,16,27)(4,17,21,25)(5,10)(6,11)(7,12)(8,15,24,22)(9,20,26,19)(14,23)(18,32)(28,30)(29,31);(1,3)(2,8)(4,18)(5,22)(6,15)(7,16)(9,23)(10,27)(11,13)(12,24)(14,21)(17,31)(19,29)(20,30)(25,28)(26,32);(1,4)(2,9)(3,14,16,18)(5,20)(6,19)(7,21)(8,23,24,32)(10,17)(11,25)(12,26)(13,28,27,31)(15,30,22,29);(1,2)(3,8)(4,17,21,25)(5,10)(6,11)(7,12)(9,20,26,19)(13,15)(14,28,18,31)(16,24)(22,27)(23,30,32,29)

Polycyclic presentation:

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

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, 289328);