#110899

order	512
nilpotency class	3
isoclinism	isoclinic class of #110892
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

594 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:

(1,2,5,9,6,10,4,7)(3,11,12,16,13,15,14,8)(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);(2,7,10,9)(8,11,16,15)(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)(4,12)(5,14)(6,13)(7,11)(9,15)(10,16)(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=efh,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18095);