#364510

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
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	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2492 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)(7,11)(8,18,20,28)(9,13)(10,14)(12,21,27,22)(15,30,19,34)(16,25)(17,26)(23,29)(24,31,35,32)(33,36);(2,4)(5,7,10,17)(6,11,14,26)(8,19)(9,16,23,33)(12,24)(13,25,29,36)(15,20)(18,30)(21,32)(22,31)(27,35)(28,34);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,28)(14,27)(16,32)(17,19)(18,29)(21,23)(25,34)(26,35)(30,36)(31,33);(2,4)(9,23)(13,29)(16,33)(18,28)(21,22)(25,36)(30,34)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,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, 271706);