#494263

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

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,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	#72264797, #72264825, #72264848, …
maximal normal subgroup of	#72264797, #72264825, #72264848, …
maximal quotient of	#72285995, #72286965, #72287172

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401459);