#410859

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257091, #72257156, #72257167
maximal normal subgroup of	#72257091, #72257156, #72257167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318055);