#354478

order	512
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280888, #72280919, #72281002, …
maximal normal subgroup of	#72280888, #72280919, #72281002, …
maximal quotient of	#72313490, #72343064, #72343108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261674);