#147476

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

704 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)(5,6,10,15,11,17,31,16)(7,18,21,45,23,46,22,40)(8,24,26,39,28,43,27,13)(9,29,32,14,33,41,61,42)(12,30,35,53,37,52,36,20)(19,47,49,66,51,65,50,44)(25,56,58,64,60,38,59,55)(34,48,54,67,63,68,62,57);(5,7,10,21,11,23,31,22)(6,12,16,36,17,37,15,35)(8,19,27,50,28,51,26,49)(9,30,32,53,33,52,61,20)(13,34,24,54,39,63,43,62)(14,40,29,46,42,45,41,18)(25,57,59,68,60,67,58,48)(38,47,55,66,56,65,64,44);(1,2,3,4)(5,8,11,28)(6,13,17,39)(7,19,23,51)(9,25,33,60)(10,27,31,26)(12,34,37,63)(14,38,42,56)(15,43,16,24)(18,44,46,66)(20,48,53,68)(21,50,22,49)(29,55,41,64)(30,57,52,67)(32,59,61,58)(35,62,36,54)(40,47,45,65)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54672);