#299366

order	512
nilpotency class	3
isoclinism	isoclinic class of #299341
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

900 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:

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

Polycyclic presentation:

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

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, 206562);