#244119

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

629 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151315);