#227618

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134814);