#351921

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, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

639 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,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, 259117);