#150962

order	512
nilpotency class	5
isoclinism	isoclinic class of #7453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1365
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂	C₂×#1365
prime radical	C₂³	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

193 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,d^2=ghi,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efg,f^b=fi,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58158);