#362533

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

879 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi

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, 269729);