#298386

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#562	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

911 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 911 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fh,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, 205582);