#429960

order	512
nilpotency class	3
isoclinism	isoclinic class of #419459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1729 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)(7,11,17,23)(8,12)(9,13)(10,14)(15,24,30,21)(16,22,31,27)(18,25)(19,26)(20,28)(29,34,36,33)(32,35);(5,7)(6,11,14,23)(8,15)(9,16,20,31)(10,17)(12,24,26,21)(13,27)(18,29,32,36)(19,30)(22,28)(25,33)(34,35);(5,8,10,19)(6,12,14,26)(7,15,17,30)(9,18,20,32)(11,21,23,24)(13,25,28,35)(16,29,31,36)(22,33,27,34);(2,4)(5,9)(6,13)(7,16,17,31)(8,18)(10,20)(11,22,23,27)(12,25)(14,28)(15,29,30,36)(19,32)(21,33,24,34)(26,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284660, #72284700, #72284833, …
maximal normal subgroup of	#72284660, #72284700, #72284833, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337156);