#147469

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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)(8,23,26,41)(9,28)(10,16)(11,33)(12,17)(13,20)(14,25,40,27)(15,31)(19,47,50,67)(21,45)(22,46)(24,39,57,65)(29,36)(30,43)(32,60)(34,42)(35,62,63,55)(37,61)(38,52)(44,49,66,51)(48,53,68,64)(54,58,59,56);(5,7,10,21,12,22,11,34)(6,13,16,36,17,38,33,37)(8,19,27,51,26,50,25,49)(9,29,30,52,32,61,31,20)(14,35,41,64,40,63,23,53)(15,42,28,18,43,45,60,46)(24,55,58,68,57,62,56,48)(39,47,59,44,65,67,54,66);(1,2,3,4)(5,8,10,27,12,26,11,25)(6,14,16,41,17,40,33,23)(7,19,21,51,22,50,34,49)(9,24,30,58,32,57,31,56)(13,35,36,64,38,63,37,53)(15,39,28,59,43,65,60,54)(18,44,45,67,46,66,42,47)(20,48,29,55,52,68,61,62)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dh,h^b=hi,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, 54665);