#229325

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,d^b=de,e^b=eh,g^b=gh,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, 136521);