#265329

order	512
nilpotency class	3
isoclinism	isoclinic class of #265327
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=i,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dh,f^c=fh

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, 172525);