#494487

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1450 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,e^2=h,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249864, #72249876, #72249897, …
maximal normal subgroup of	#72249864, #72249876, #72249897, …
maximal quotient of	#72265731, #72280693, #72285180

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401683);