#147061

order	512
nilpotency class	4
isoclinism	isoclinic class of #11669
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
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

538 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)(7,18)(8,23,25,14)(9,28)(10,17)(11,34)(12,16)(13,20)(15,32)(19,45,49,64)(21,47)(22,46)(24,55,30,63)(26,54,27,41)(29,37)(31,44)(33,60)(35,42)(36,48,53,51)(38,62)(39,52)(40,58,43,57)(50,67,59,68)(56,66,61,65);(5,7,10,22,12,21,11,35)(6,13,17,37,16,39,34,38)(8,19,26,50,25,49,27,59)(9,29,31,52,33,62,32,20)(14,36,41,65,23,53,54,66)(15,42,28,18,44,46,60,47)(24,56,57,51,30,61,58,48)(40,67,63,64,43,68,55,45);(1,2,3,4)(5,8,11,27,12,25,10,26)(6,14,34,54,16,23,17,41)(7,19,35,59,21,49,22,50)(9,30,32,57,33,24,31,58)(13,36,38,66,39,53,37,65)(15,43,60,63,44,40,28,55)(18,45,42,68,47,64,46,67)(20,51,62,56,52,48,29,61)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54257);