#212910

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 120106);