#351901

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

726 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^c=dh,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259097);