#299536

order	512
nilpotency class	3
isoclinism	isoclinic class of #299505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3498	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

968 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,10,12,7)(4,14,9,6)(11,16,15,13)(17,18,23,29)(19,30,20,25)(21,36,38,44)(22,28,34,27)(24,33,40,32)(26,42,45,37)(31,46,35,39)(41,47,43,48);(1,3)(2,7)(4,15)(5,12)(6,13)(8,10)(9,11)(14,16)(17,19,22,32,23,20,34,33)(18,24,27,30,29,40,28,25)(21,31,37,47,38,35,42,48)(26,39,36,43,45,46,44,41);(1,4,6,2,5,9,14,8)(3,11,13,10,12,15,16,7)(17,20)(18,25)(19,23)(21,31)(22,33)(24,27)(26,43)(28,40)(29,30)(32,34)(35,38)(36,46)(37,47)(39,44)(41,45)(42,48);(2,8)(3,12)(4,9)(13,16)(17,21,22,37,23,38,34,42)(18,26,27,36,29,45,28,44)(19,31,32,47,20,35,33,48)(24,39,30,43,40,46,25,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,g^b=gh,d^c=dhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206732);