#371122

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

894 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)(7,15,18,11)(8,12)(9,14)(10,13)(16,19,28,23)(17,25,32,29)(20,21)(22,26)(24,27)(30,36,31,33)(34,35);(1,3,4,2)(5,6)(7,16,18,28)(8,20)(9,14)(10,13)(11,19,15,23)(12,21)(17,31,32,30)(22,34)(24,27)(25,36,29,33)(26,35);(5,7)(6,11)(8,17)(9,23)(10,18)(12,25)(13,15)(14,28)(16,27)(19,24)(20,29)(21,32)(22,36)(26,31)(30,34)(33,35);(5,8,9,22,10,21,24,35)(6,12,14,26,13,20,27,34)(7,17,23,36,18,32,19,33)(11,25,28,31,15,29,16,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=eg,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,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, 278318);