#498935

order	512
nilpotency class	4
isoclinism	isoclinic class of #498886
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2137 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)(5,6,11,8)(7,15,19,28)(9,20,23,13)(10,21,25,14)(12,24,16,18)(17,31,33,29)(22,27,36,34)(26,32,30,35);(5,7)(6,12)(8,16)(9,17)(10,24)(11,19)(13,26)(14,28)(15,21)(18,25)(20,30)(22,35)(23,33)(27,31)(29,34)(32,36);(2,4)(5,8,11,6)(7,16,19,12)(9,13,23,20)(10,14,25,21)(15,18,28,24)(17,26,33,30)(22,34,36,27)(29,35,31,32);(5,9,11,23)(6,13,8,20)(7,17,19,33)(10,22,25,36)(12,26,16,30)(14,27,21,34)(15,29,28,31)(18,32,24,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256645, #72256647
maximal normal subgroup of	#72256645, #72256647
maximal quotient of	#72316498

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406131);