#148665

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321196, #72321284, #72321353, …
maximal normal subgroup of	#72321196, #72321284, #72321353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55861);