#244137

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

615 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)(3,10,12,15,11,16,13,6)(17,18,21,25,22,26,32,39)(19,27,29,36,30,42,35,23)(20,24,33,40,34,41,47,48)(28,43,44,38,31,45,46,37);(1,3,5,11)(2,6,9,15)(4,13,14,12)(7,10,8,16)(17,19,22,30)(18,23,26,36)(20,31,34,28)(21,35,32,29)(24,38,41,37)(25,42,39,27)(33,44,47,46)(40,43,48,45);(2,7)(4,14)(6,10)(8,9)(12,13)(15,16)(23,36)(27,42)(37,38)(43,45);(3,11)(6,15)(10,16)(12,13)(17,20,32,47,22,34,21,33)(18,24,39,48,26,41,25,40)(19,28,29,44,30,31,35,46)(23,37,27,43,36,38,42,45)

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,d^b=dh,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151333);