#147102

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(2,4)(5,6,10,16,12,17,11,34)(7,18,22,46,24,37,23,54)(8,25,28,57,30,58,29,61)(9,31,32,43,33,62,26,15)(13,35,38,21,39,53,45,20)(14,40,41,60,42,59,56,27)(19,47,50,64,52,66,51,36)(44,65,67,48,55,63,68,49);(5,7,9,21,10,22,32,53,12,24,33,20,11,23,26,35)(6,13,15,37,34,45,62,46,17,39,43,18,16,38,31,54)(8,19,27,49,28,50,40,65,30,52,60,48,29,51,59,63)(14,36,25,55,56,66,61,67,42,64,58,44,41,47,57,68);(1,2,3,4)(5,8,26,59,11,29,33,60,12,30,32,40,10,28,9,27)(6,14,31,57,16,41,43,58,17,42,62,61,34,56,15,25)(7,19,35,63,23,51,20,48,24,52,53,65,22,50,21,49)(13,36,54,68,38,47,18,44,39,64,46,67,45,66,37,55)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=hi,e^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54298);