#299821

order	512
nilpotency class	3
isoclinism	isoclinic class of #299805
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	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

828 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)(3,12)(4,13)(5,11)(6,10)(7,9)(8,14)(15,16)(17,18)(19,30)(20,25)(21,36)(22,28)(23,29)(24,32)(26,37)(27,34)(31,45)(33,40)(35,39)(38,42)(41,47)(43,48)(44,46);(1,3)(2,8)(4,9)(5,13)(6,14)(7,16)(10,12)(11,15)(17,19,22,32,23,20,34,33)(18,24,27,30,29,40,28,25)(21,31,37,46,38,35,48,47)(26,39,36,44,43,45,42,41);(1,4,7,12,6,15,5,8)(2,3,11,13,10,14,9,16)(17,20)(18,25)(19,23)(21,35)(22,33)(24,27)(26,41)(28,40)(29,30)(31,38)(32,34)(36,39)(37,47)(42,45)(43,44)(46,48);(1,5,6,7)(2,9,10,11)(3,13,14,16)(4,12,15,8)(17,21,22,37,23,38,34,48)(18,26,27,36,29,43,28,42)(19,31,32,46,20,35,33,47)(24,39,30,44,40,45,25,41)

Polycyclic presentation:

2^9|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,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207017);