#427261

order	512
nilpotency class	3
isoclinism	isoclinic class of #417542
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1532 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)(8,12,19,26)(9,13,21,28)(15,29,31,22)(16,27,20,23)(18,25,32,35)(30,34,33,36);(5,7)(6,11)(8,15)(9,20)(10,17)(12,22)(13,27)(14,24)(16,21)(18,30)(19,31)(23,28)(25,34)(26,29)(32,33)(35,36);(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,25)(14,26)(16,30)(17,31)(20,33)(21,32)(23,34)(24,29)(27,36)(28,35);(2,4)(5,9)(6,13)(7,16,17,20)(8,18)(10,21)(11,23,24,27)(12,25)(14,28)(15,30,31,33)(19,32)(22,34,29,36)(26,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251368, #72251390, #72251395, …
maximal normal subgroup of	#72251368, #72251390, #72251395, …
maximal quotient of	#72251439

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334457);