#379513

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

904 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)(5,6,11,15)(7,16,20,33)(8,13,22,30)(9,23,21,36)(10,25,24,14)(12,26,29,19)(17,32,35,31)(18,27,34,28);(5,7)(6,12,15,29)(8,17)(9,18)(10,19)(11,20)(13,28,30,27)(14,16,25,33)(21,34)(22,35)(23,32,36,31)(24,26);(5,8,10,21,11,22,24,9)(6,13,25,36,15,30,14,23)(7,17,19,34,20,35,26,18)(12,27,33,32,29,28,16,31);(1,3,4,2)(5,9,24,22,11,21,10,8)(6,13,25,36,15,30,14,23)(7,18,26,35,20,34,19,17)(12,28,33,31,29,27,16,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286709);