#495233

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fh,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,e^b=ei,g^b=gi,d^c=di

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, 402429);