#232494

order	512
nilpotency class	3
isoclinism	isoclinic class of #232487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1255 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 1255 conjugacy classes of subgroups — too many to draw. Only its 196 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,23,28)(19,29,33,40)(20,34,38,43)(21,26,22,27)(24,32,41,31)(25,36,44,37)(30,46,47,39)(35,45,48,42);(4,11)(6,12)(7,14)(9,15)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,42)(26,29)(27,40)(28,41)(30,36)(34,39)(37,47)(38,48)(43,46)(44,45);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20,23,38)(18,25,28,44)(19,30,33,47)(21,36,22,37)(24,39,41,46)(26,34,27,43)(29,45,40,42)(31,35,32,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,40,41,29)(25,34,44,43)(30,35,47,48)(39,42,46,45)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,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, 139690);