#424968

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 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)(7,19)(8,23)(9,25,27,15)(10,17)(11,16)(12,18)(13,21)(14,22)(20,30,34,33)(24,29,35,36)(26,31,28,32);(1,2)(3,4)(5,7,10,21,12,22,11,8)(6,13,16,19,18,23,17,14)(9,24,26,34,27,35,28,20)(15,30,31,36,25,33,32,29);(2,4)(5,8,11,22,12,21,10,7)(6,14,17,23,18,19,16,13)(9,24,26,34,27,35,28,20)(15,30,31,36,25,33,32,29);(5,9)(6,15)(7,20)(8,24)(10,28)(11,26)(12,27)(13,29)(14,30)(16,32)(17,31)(18,25)(19,33)(21,35)(22,34)(23,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332164);