#368237

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1435 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:

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

Polycyclic presentation:

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

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, 275433);