#303949

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

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

normal subgroups only

This group has 893 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,14,12)(17,18)(19,27)(20,22)(21,25)(23,26)(24,28)(29,32)(30,31);(1,3,5,4)(2,6,8,7)(11,14)(12,16)(17,19,21,28)(18,24,25,27)(20,30,26,32)(22,29,23,31);(1,3,5,4)(2,7,8,6)(9,11,12,10,13,14,16,15)(17,20,23,25,21,26,22,18)(19,24,31,30,28,27,29,32);(1,4,5,3)(2,7,8,6)(18,25)(20,26)(24,27)(30,32)

Polycyclic presentation:

2^9|a^2=i,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, 211145);