#449579

order	512
nilpotency class	3
isoclinism	isoclinic class of #417538
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2190 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)(7,11)(8,12)(9,13)(10,14)(15,21)(16,22)(17,23)(18,31)(19,26)(20,27)(24,29)(25,33)(28,36)(30,35)(32,34);(5,7,10,17)(6,11,14,23)(8,15,19,29)(9,16,20,30)(12,24,26,21)(13,22,27,35)(18,32,33,28)(25,34,31,36);(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,25)(14,26)(16,28)(17,29)(20,33)(22,34)(23,24)(27,31)(30,32)(35,36);(2,4)(5,9)(6,13)(7,16)(8,18)(10,20)(11,22)(12,25)(14,27)(15,28)(17,30)(19,33)(21,34)(23,35)(24,36)(26,31)(29,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252226, #72284660, #72284700, …
maximal normal subgroup of	#72252226, #72284660, #72284700, …
maximal quotient of	#72252219

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356775);