#269201

order	512
nilpotency class	3
isoclinism	isoclinic class of #269147
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	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

784 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=h,d^2=i,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, 176397);