#189115

order	512
nilpotency class	3
isoclinism	isoclinic class of #19279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^c=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, 96311);