#494488

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×#3060	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1386 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)(5,6,11,18,20,21,10,31,12,17,34,35,9,15,30,16)(7,19,25,8,22,28,36,26,24,27,32,13,14,33,23,29);(1,2,3,4)(5,7,20,14,12,24,9,22)(6,13,15,29,17,8,21,26)(10,32,34,36,30,25,11,23)(16,33,35,19,31,28,18,27);(5,8)(6,14)(7,21)(9,29)(10,19)(11,33)(12,13)(15,24)(16,25)(17,22)(18,36)(20,26)(23,35)(27,30)(28,34)(31,32);(5,9,12,20)(6,15,17,21)(7,22,24,14)(8,26,13,29)(10,11,30,34)(16,35,31,18)(19,28,27,33)(23,25,36,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,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	#72250045, #72250051, #72250069, …
maximal normal subgroup of	#72250045, #72250051, #72250069, …
maximal quotient of	#72265420

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401684);