#121962

order	512
nilpotency class	3
isoclinism	isoclinic class of #121960
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

355 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,12)(4,17,21,27)(5,22)(6,10)(7,11)(8,14)(9,20,28,19)(13,29,31,23)(15,25)(16,24)(18,32,30,26)(33,34,38,44,39,43,37,42)(35,40,47,56,48,55,46,54)(36,49,52,41,50,59,51,58)(45,60,63,53,61,64,62,57);(1,3)(2,8,11,25)(4,18)(5,14)(6,15)(7,16)(9,26,28,32)(10,12,22,24)(13,19)(17,29,27,23)(20,31)(21,30)(33,35,37,46,39,48,38,47)(34,40,42,54,43,55,44,56)(36,45,52,63,50,61,51,62)(41,57,59,60,58,53,49,64);(1,4)(2,9,11,28)(3,13)(5,19)(6,20)(7,21)(8,23,25,29)(10,17,22,27)(12,26,24,32)(14,30)(15,18)(16,31)(33,36,37,52,39,50,38,51)(34,41,42,59,43,58,44,49)(35,45,46,63,48,61,47,62)(40,53,54,64,55,57,56,60)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=ei,f^c=fh,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, 29158);