#351857

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
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

735 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)(5,6,9,12)(7,13,16,20)(8,17,14,11)(10,18,19,15)(21,22)(23,32)(24,34)(25,29)(26,31)(27,28)(30,33)(35,36);(5,7,14,18,9,16,8,15)(6,10,17,20,12,19,11,13)(21,23,26,27)(22,28,31,32)(24,35,25,33)(29,36,34,30);(1,2,3,4)(21,24,26,25)(22,29,31,34)(23,33,27,35)(28,30,32,36);(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=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,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259053);