#298383

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×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1031 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1031 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,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, 205579);