#303936

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

719 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,17)(11,18,21,15)(12,23,16,13)(19,22,24,20)(25,26)(27,31)(28,40)(29,34)(30,35)(32,44)(33,42)(36,53)(37,51)(38,47)(39,48)(41,46)(43,52)(45,50)(49,54)(55,56);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,24)(13,20)(14,21)(16,19)(17,18)(22,23)(25,27,30,39)(26,31,35,48)(28,41,43,55)(29,37,42,38)(32,49,50,53)(33,47,34,51)(36,44,54,45)(40,46,52,56);(1,3,5,4)(2,7,8,6)(9,12,13,10,14,16,23,17)(11,19,22,15,21,24,20,18)(25,28,42,45,30,43,29,44)(26,32,34,52,35,50,33,40)(27,36,38,55,39,54,37,41)(31,46,51,49,48,56,47,53);(1,4,5,3)(2,7,8,6)(9,13,14,23)(10,16,17,12)(11,20,21,22)(15,19,18,24)(28,43)(32,50)(36,54)(40,52)(41,55)(44,45)(46,56)(49,53)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=efh,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211132);