#269152

order	512
nilpotency class	3
isoclinism	isoclinic class of #269147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60810	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176348);