#212913

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
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	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

714 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)(7,13)(8,15)(10,16)(12,18)(14,19)(17,20)(21,22)(23,30)(24,25)(26,34)(27,29)(28,32)(31,35)(33,36);(5,6,9,11)(7,10,14,17)(8,12,15,18)(13,16,19,20)(21,23,25,31,27,33,26,32)(22,28,34,36,29,35,24,30);(5,7,8,13,9,14,15,19)(6,10,12,16,11,17,18,20)(21,24,25,22,27,34,26,29)(23,30,31,28,33,36,32,35);(1,3,4,2)(6,11)(10,17)(12,18)(16,20)(21,25,27,26)(22,24,29,34)(23,31,33,32)(28,30,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fh,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

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, 120109);