#496380

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

882 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403576);