#411497

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

928 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=e,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916, #72249848, #72252980, …
maximal normal subgroup of	#72224916, #72249848, #72252980, …
maximal quotient of	#72224913

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318693);