#229317

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, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=hi,d^2=eh,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, 136513);