#200127

order	512
nilpotency class	3
isoclinism	isoclinic class of #200122
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3219	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1075 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)(3,5)(4,12,14,8)(6,11)(7,9)(10,13,16,15)(17,18)(19,23,28,37)(20,30,31,42)(21,26)(22,25)(24,32,39,34)(27,43)(29,38,33,41)(35,40)(36,46)(44,47)(45,48);(1,3,5,2,7,11,6,9)(4,10,13,8,14,16,15,12)(17,19,22,28)(18,23)(20,27,31,44)(21,33,35,29)(24,36)(25,37)(26,41)(30,47)(32,46,34,45)(38,40)(39,48)(42,43);(1,4)(2,8)(3,10)(5,13)(6,15)(7,14)(9,12)(11,16)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,39)(26,42)(28,44)(29,46)(30,40)(32,35)(33,45)(37,48)(38,47)(41,43);(1,5,7,6)(2,3,9,11)(4,13,14,15)(8,10,12,16)(19,28)(23,37)(27,44)(29,33)(36,48)(38,41)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107323);