#333209

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18,22,28,24,30,23,29)(19,31,33,44,35,43,34,25)(20,26,37,46,38,42,41,47)(21,39,40,27,32,45,36,48);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(17,19,24,35)(18,25,30,44)(20,36,38,40)(21,37,32,41)(22,34,23,33)(26,45,42,39)(27,47,48,46)(28,43,29,31);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,20)(18,26)(19,32)(21,35)(22,37)(23,41)(24,38)(25,27)(28,46)(29,47)(30,42)(31,45)(33,36)(34,40)(39,43)(44,48);(1,3)(2,8)(4,14)(5,12)(6,13)(7,11)(9,15)(10,16)(17,21,24,32)(18,27,30,48)(19,20,35,38)(22,36,23,40)(25,42,44,26)(28,39,29,45)(31,46,43,47)(33,41,34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240405);