#405275

order	512
nilpotency class	3
isoclinism	isoclinic class of #394273
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

2410 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)(5,6,10,14)(7,11,17,24)(8,12,20,27)(9,13,21,28)(15,25,18,22)(16,23,30,35)(19,32,33,26)(29,34,31,36);(2,4)(5,7)(6,11)(8,18,20,15)(9,16)(10,17)(12,25,27,22)(13,23)(14,24)(19,31,33,29)(21,30)(26,36,32,34)(28,35);(5,8)(6,12)(7,15)(9,19)(10,20)(11,22)(13,26)(14,27)(16,29)(17,18)(21,33)(23,34)(24,25)(28,32)(30,31)(35,36);(5,9)(6,13)(7,16)(8,19)(10,21)(11,23)(12,26)(14,28)(15,29)(17,30)(18,31)(20,33)(22,34)(24,35)(25,36)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252226, #72258432, #72258442, …
maximal normal subgroup of	#72252226, #72258432, #72258442, …
maximal quotient of	#72252221, #72258959, #72266204

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312471);