#427935

order	512
nilpotency class	3
isoclinism	isoclinic class of #427935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15376
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,8)(5,16,18,9)(6,10)(11,22)(12,20,17,25)(13,21)(14,23,29,30)(15,24)(19,27)(26,31,28,32);(1,3)(2,7)(4,11)(5,17)(6,13)(8,22)(9,25)(10,21)(12,18)(14,28)(15,27)(16,20)(19,24)(23,31)(26,29)(30,32);(1,4)(2,8)(3,11)(5,14)(6,15)(7,19)(9,23)(10,24)(12,26)(13,27)(16,30)(17,28)(18,29)(20,31)(21,22)(25,32);(1,5)(2,9)(3,12,13,17)(4,14)(6,18)(7,20,21,25)(8,23)(10,16)(11,26,27,28)(15,29)(19,31,22,32)(24,30)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237754, #72237759, #72237763, …
maximal normal subgroup of	#72237754, #72237759, #72237763, …
maximal quotient of	#72265862, #72265865

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335131);