#343135

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	H(3,𝔽₄)
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

595 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250331);