#368075

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1642 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)(5,6,10,14)(7,15,18,11)(8,12,19,25)(9,21,22,13)(16,23,30,27)(17,24,31,28)(20,33,32,26)(29,35,36,34);(1,3,4,2)(7,16)(9,22)(11,23)(13,21)(15,27)(17,29)(18,30)(20,32)(24,34)(26,33)(28,35)(31,36);(5,7,9,16,10,18,22,30)(6,11,13,23,14,15,21,27)(8,17,20,29,19,31,32,36)(12,24,26,34,25,28,33,35);(5,8)(6,12)(7,17)(9,20)(10,19)(11,24)(13,26)(14,25)(15,28)(16,29)(18,31)(21,33)(22,32)(23,34)(27,35)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,g^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,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, 275271);