#115363

order	512
nilpotency class	3
isoclinism	isoclinic class of #115356
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

771 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22559);