#233766

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1178 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1178 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(5,6,11,16)(7,17,8,19)(9,15,10,14)(12,20,13,18)(21,22,27,31)(23,32,35,42)(24,29,38,45)(25,30,26,39)(28,40,43,34)(33,48,50,47)(36,44,37,46)(41,51,52,49);(1,2)(3,4)(5,7)(6,12)(8,11)(9,20)(10,18)(13,16)(14,19)(15,17)(21,23,27,35)(22,28,31,43)(24,33,38,50)(25,34,26,40)(29,41,45,52)(30,42,39,32)(36,49,37,51)(44,47,46,48);(2,4)(5,8,11,7)(6,13,16,12)(9,20,10,18)(14,19,15,17)(21,24,27,38)(22,29,31,45)(23,33,35,50)(25,37,26,36)(28,41,43,52)(30,46,39,44)(32,47,42,48)(34,51,40,49);(7,8)(12,13)(17,19)(18,20)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,36,38,37)(28,42,43,32)(29,44,45,46)(33,49,50,51)(41,47,52,48)

Polycyclic presentation:

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

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, 140962);