#343118

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	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 639 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=h,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, 250314);