#494262

order	512
nilpotency class	4
isoclinism	isoclinic class of #483425
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1046 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)(6,11,13,24)(8,18)(9,21)(12,23,25,34)(14,27,22,19)(16,30)(17,29)(20,33)(26,35,32,31)(28,36);(5,6,9,22,10,13,21,14)(7,12,17,32,15,25,29,26)(8,19,18,11,20,27,33,24)(16,31,30,23,28,35,36,34);(2,4)(5,7)(6,12)(8,16)(9,17)(10,15)(11,23)(13,25)(14,26)(18,30)(19,31)(20,28)(21,29)(22,32)(24,34)(27,35)(33,36);(7,15)(12,25)(16,28)(17,29)(23,34)(26,32)(30,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225127, #72239487, #72249876, …
maximal normal subgroup of	#72225127, #72239487, #72249876, …
maximal quotient of	#72224980, #72286277, #72287172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401458);