#147545

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231488, #72231501, #72231505, …
maximal normal subgroup of	#72231488, #72231501, #72231505, …
maximal quotient of	#72293221, #72293282

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54741);