#221207

order	512
nilpotency class	3
isoclinism	isoclinic class of #221204
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	#3497	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

869 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)(9,12)(10,14)(11,15)(13,16)(17,18)(19,28)(20,32,35,42)(21,24)(22,37)(23,27)(25,30)(26,34,33,39)(29,44,47,40)(31,41)(36,38)(43,46,48,45);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,13)(12,16)(17,19,24,31)(18,25,21,36)(20,29,35,47)(22,38,23,30)(26,40,33,44)(27,41,37,28)(32,43,42,48)(34,46,39,45);(2,7)(3,10)(4,12)(6,13)(8,9)(14,15)(17,20)(18,26)(19,29)(21,33)(22,34)(23,39)(24,35)(25,40)(27,42)(28,43)(30,45)(31,47)(32,37)(36,44)(38,46)(41,48);(2,8)(6,14)(7,9)(13,15)(17,21,24,18)(19,28,31,41)(20,33,35,26)(22,27,23,37)(25,38,36,30)(29,43,47,48)(32,34,42,39)(40,46,44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=dhi,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128403);