#232172

order	512
nilpotency class	3
isoclinism	isoclinic class of #232169
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ef,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=eh,c^b=cef,d^b=de,f^b=fh,g^b=gh,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139368);