#364195

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1506 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=i,d^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,f^c=fi,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, 271391);