#226992

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#537	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,d^b=dei,e^b=eh,d^c=di,f^c=fh,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, 134188);