#369180

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi

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, 276376);