#327971

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,22)(20,28)(21,25)(23,26)(24,30)(27,32)(29,31)(33,34)(35,42)(36,40)(37,45)(38,41)(39,43)(44,48)(46,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,18)(20,29,31,28)(21,25,23,26)(24,32,27,30)(36,37)(39,47)(40,45)(43,46);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,22,31)(18,24,19,27)(21,32,23,30)(25,28,26,29)(33,35,38,44)(34,39,41,47)(36,43,37,46)(40,48,45,42);(4,12)(7,14)(10,16)(13,15)(17,21,22,23)(18,25,19,26)(20,30,31,32)(24,29,27,28)(33,36,38,37)(34,40,41,45)(35,43,44,46)(39,48,47,42)

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=de,f^c=fh,g^c=ghi

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, 235167);