#380370

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,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, 287566);