#415184

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 802 conjugacy classes of subgroups — too many to draw. Only its 242 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,29,33,41)(20,34,38,44)(21,28,22,27)(24,32,40,31)(25,36,43,37)(30,39,46,47)(35,42,45,48);(2,5)(6,8)(17,19,21,31,23,33,22,32)(18,24,27,29,26,40,28,41)(20,35,36,30,38,45,37,46)(25,42,44,47,43,48,34,39);(1,3,4,7)(2,6,5,8)(17,20,21,36,23,38,22,37)(18,25,27,44,26,43,28,34)(19,30,31,45,33,46,32,35)(24,39,29,42,40,47,41,48);(9,11,12,16)(10,13,14,15)(18,26)(24,40)(25,43)(27,28)(29,41)(34,44)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322380);