#364530

order	512
nilpotency class	3
isoclinism	isoclinic class of #23990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2395 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)(7,15,18,11)(8,12)(9,21)(10,13)(14,23)(16,24,22,28)(17,25,31,29)(19,27)(20,33)(26,32)(30,36,34,35);(2,4)(7,16,18,22)(11,24,15,28)(17,30,31,34)(25,35,29,36);(5,7)(6,11)(8,17)(9,22)(10,18)(12,25)(13,15)(14,28)(16,23)(19,31)(20,34)(21,24)(26,36)(27,29)(30,32)(33,35);(2,4)(5,8,10,19)(6,12,13,27)(7,17,18,31)(9,20,23,32)(11,25,15,29)(14,26,21,33)(16,30,22,34)(24,35,28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,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,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, 271726);