#221332

order	512
nilpotency class	3
isoclinism	isoclinic class of #221184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61492	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fhi,g^c=gh,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128528);