#200719

order	512
nilpotency class	3
isoclinism	isoclinic class of #200710
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#60693	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1023 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286642, #72286759, #72286809
maximal normal subgroup of	#72286642, #72286759, #72286809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107915);