#494271

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1044 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)(7,16,19,34)(8,13)(9,22)(10,20)(11,15)(12,18,29,32)(14,25)(17,33,27,31)(21,24)(23,30,36,28)(26,35);(5,7,10,27,11,19,8,17)(6,12,13,28,15,29,20,30)(9,23,25,18,26,36,21,32)(14,31,22,16,24,33,35,34);(2,4)(5,6)(7,12)(8,20)(9,24)(10,13)(11,15)(14,26)(16,32)(17,30)(18,34)(19,29)(21,22)(23,33)(25,35)(27,28)(31,36);(5,8,11,10)(6,13,15,20)(7,17,19,27)(9,21,26,25)(12,28,29,30)(14,22,24,35)(16,33,34,31)(18,23,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250027, #72250051, #72250078, …
maximal normal subgroup of	#72250027, #72250051, #72250078, …
maximal quotient of	#72265366

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401467);