#232613

order	512
nilpotency class	3
isoclinism	isoclinic class of #232606
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1322 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

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, 139809);