#303933

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 771 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=ef,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, 211129);