#410863

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

1157 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)(7,13)(8,22,24,36)(9,15,27,34)(10,16)(11,30)(12,17)(14,25,19,26)(18,28,35,29)(20,32)(21,33)(23,31);(5,7,10,20,12,21,11,31)(6,13,16,32,17,33,30,23)(8,15,25,35,24,34,26,18)(9,19,28,36,27,14,29,22);(2,4)(5,8,11,26,12,24,10,25)(6,14,16,22,17,19,30,36)(7,18,20,15,21,35,31,34)(9,23,29,33,27,32,28,13);(5,9)(6,15)(7,19)(8,23)(10,28)(11,29)(12,27)(13,25)(14,21)(16,35)(17,34)(18,30)(20,36)(22,31)(24,32)(26,33)

Polycyclic presentation:

2^9|a^2=e,b^2=gh,c^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,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	#72256961, #72257156, #72257214
maximal normal subgroup of	#72256961, #72257156, #72257214

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318059);