#419460

order	512
nilpotency class	3
isoclinism	isoclinic class of #419459
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

1525 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:

(5,6)(7,15,18,11)(8,12,20,26)(9,13)(10,14)(16,22)(17,23,31,27)(19,25,32,35)(21,28)(24,30)(29,34)(33,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,27)(14,15)(19,29)(20,30)(21,31)(22,26)(23,28)(25,33)(32,36)(34,35);(2,4)(5,8)(6,12,14,26)(7,16)(9,19)(10,20)(11,22,15,24)(13,25,28,35)(17,29)(18,30)(21,32)(23,33,27,34)(31,36);(5,9,10,21)(6,13,14,28)(7,17,18,31)(8,19,20,32)(11,23,15,27)(12,25,26,35)(16,29,30,36)(22,33,24,34)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284700, #72284731, #72284833, …
maximal normal subgroup of	#72284700, #72284731, #72284833, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326656);