#225587

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

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

607 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fi,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,e^d=eh

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, 132783);