#110874

order	512
nilpotency class	3
isoclinism	isoclinic class of #110864
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,7,4,14)(3,10,13,15,11,16,12,8)(17,18)(19,27)(20,31)(21,34)(22,26)(23,29)(24,32)(25,35)(28,41)(30,37)(33,46)(36,48)(38,47)(39,43)(40,44)(42,45);(2,7)(4,5)(8,15)(12,13)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,36,38,39)(25,37,34,40)(31,45,46,41)(32,43)(33,42)(35,44)(47,48);(1,3)(2,8,7,15)(4,12)(5,13)(6,11)(9,10,14,16)(17,20,21,32,22,33,35,47)(18,24,25,31,26,38,34,46)(19,28,30,43,29,42,44,48)(23,36,37,45,27,39,40,41)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18070);