#494725

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2000 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)(7,16,18,32)(8,12)(9,21)(10,14,26,30)(11,15)(13,24)(17,33,35,31)(19,28)(20,25)(22,27)(23,34,36,29);(5,7,10,25)(6,8,14,16)(9,17,23,28)(11,18,26,12)(13,19,29,31)(15,20,30,32)(21,22,34,33)(24,35,36,27);(2,4)(5,8,11,20)(6,12,15,25)(7,14,18,30)(9,22,24,19)(10,16,26,32)(13,28,21,27)(17,34,35,29)(23,33,36,31);(5,9,11,24)(6,13,15,21)(7,17,18,35)(8,19,20,22)(10,23,26,36)(12,27,25,28)(14,29,30,34)(16,31,32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,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	#72312409, #72312417, #72312419, …
maximal normal subgroup of	#72312409, #72312417, #72312419, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401921);