#347172

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62869	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2003 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gh,f^d=fh

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, 254368);