#461462

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

954 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368658);