#473270

order	512
nilpotency class	3
isoclinism	isoclinic class of #455822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 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)(8,13,22,29)(9,25)(10,14)(11,15)(12,18)(17,32,21,35)(19,27)(20,31)(23,36,24,30)(26,34,28,33);(1,2,3,4)(5,6)(7,16)(8,13)(9,14)(10,25)(11,15)(12,19)(17,32)(18,27)(20,31)(21,35)(22,29)(23,30)(24,36)(26,34)(28,33);(2,4)(5,7,11,20)(6,12,15,27)(8,21,22,17)(9,19,10,18)(13,28,29,26)(14,31,25,16)(23,33,24,34)(30,32,36,35);(5,8,9,23,11,22,10,24)(6,13,14,30,15,29,25,36)(7,17,18,33,20,21,19,34)(12,26,16,32,27,28,31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380466);