#331634

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 154 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)(17,18,20,25,21,24,28,23)(19,22,27,30,26,31,32,29)(33,34)(35,38)(36,40)(37,39)(41,45)(42,47)(43,46)(44,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,26)(18,22,24,31)(20,27,28,32)(23,29,25,30)(33,35,36,42,37,43,44,41)(34,38,40,47,39,46,48,45);(4,12)(7,14)(10,16)(13,15)(18,23)(20,28)(22,29)(24,25)(27,32)(30,31)(35,41)(36,44)(38,45)(40,48)(42,43)(46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(19,26)(23,25)(27,32)(34,39)(38,46)(40,48)(45,47)

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238830);