#232903

order	512
nilpotency class	3
isoclinism	isoclinic class of #232897
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	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1136 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 190 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)(3,12)(4,13)(5,16)(6,11)(7,10)(8,9)(14,15)(17,18,23,28)(19,29,33,39)(20,34,37,42)(21,27,22,26)(24,31,40,32)(25,35,43,36)(30,38,47,41)(44,45,48,46);(1,3)(2,8)(4,11)(5,13)(6,14)(7,9)(10,15)(12,16)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,41)(26,39)(27,29)(28,40)(34,44)(35,45)(36,46)(37,47)(38,43)(42,48);(1,4,6,8)(2,9,11,13)(3,10,14,16)(5,15,7,12)(17,20,23,37)(18,25,28,43)(19,30,33,47)(21,36,22,35)(24,38,40,41)(26,34,27,42)(29,44,39,48)(31,46,32,45);(1,5,6,7)(2,10,11,16)(3,13,14,9)(4,12,8,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,39,40,29)(25,42,43,34)(30,45,47,46)(38,44,41,48)

Polycyclic presentation:

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

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, 140099);