#115069

order	512
nilpotency class	3
isoclinism	isoclinic class of #115065
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60917	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

770 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=ehi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22265);