#268079

order	512
nilpotency class	3
isoclinism	isoclinic class of #268043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

779 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi,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, 175275);