#147420

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

698 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)(7,18,21,46,24,38,23,54)(8,25,27,42,28,41,33,14)(9,29,30,43,32,60,31,15)(13,35,37,22,39,52,45,20)(19,47,49,63,51,67,53,44)(26,40,57,56,58,66,61,59)(36,48,55,68,65,50,64,62);(5,7,9,22,10,21,30,52,12,24,32,20,11,23,31,35)(6,13,15,38,34,45,60,46,17,39,43,18,16,37,29,54)(8,19,26,50,33,53,61,68,28,51,58,48,27,49,57,62)(14,36,40,63,25,55,56,67,42,65,66,44,41,64,59,47);(1,2,3,4)(5,8,12,28)(6,14,17,42)(7,19,24,51)(9,26,32,58)(10,33,11,27)(13,36,39,65)(15,40,43,66)(16,41,34,25)(18,44,38,63)(20,48,22,50)(21,53,23,49)(29,59,60,56)(30,61,31,57)(35,62,52,68)(37,64,45,55)(46,67,54,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di,e^c=ei,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, 54616);