#150485

order	512
nilpotency class	5
isoclinism	isoclinic class of #150484
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11775
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2930 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

283 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 52 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=g,d^2=gi,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efg,g^b=gi,d^c=dhi,f^c=fi,e^d=eh,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57681);