#381885

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6)(7,17,21,32)(8,13)(9,14)(10,15)(11,16)(12,20,31,26)(18,19,22,34)(23,24)(25,27)(28,33)(29,35,36,30);(5,7)(6,12)(8,22)(9,19)(10,26)(11,21)(13,29)(14,30)(15,32)(16,31)(17,33)(18,23)(20,28)(24,36)(25,34)(27,35);(2,4)(5,8,11,23)(6,13,16,24)(7,18,21,22)(9,15,25,33)(10,27,28,14)(12,29,31,36)(17,19,32,34)(20,35,26,30);(5,9,10,24,11,25,28,13)(6,14,15,23,16,27,33,8)(7,19,26,36,21,34,20,29)(12,30,32,18,31,35,17,22)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=efh,e^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,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, 289081);