#305021

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

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

normal subgroups only

This group has 729 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) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=ef,d^2=ehi,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, 212217);