#494684

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#35659	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2039 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)(7,17,21,26)(8,13)(9,14)(10,27,18,15)(11,16)(12,20)(19,32,35,36)(22,28)(23,34,33,29)(24,30)(25,31);(5,7,18,20,11,21,10,14)(6,12,27,26,16,9,15,17)(8,19,33,28,24,35,23,31)(13,22,34,36,30,25,29,32);(2,4)(5,8,11,24)(6,13,16,30)(7,19,21,35)(9,25,12,22)(10,23,18,33)(14,31,20,28)(15,29,27,34)(17,32,26,36);(5,9,11,12)(6,14,16,20)(7,15,21,27)(8,22,24,25)(10,26,18,17)(13,28,30,31)(19,34,35,29)(23,32,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303176, #72306342, #72306350, …
maximal normal subgroup of	#72303176, #72306342, #72306350, …
maximal quotient of	#72313472

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401880);