#150176

order	512
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1343
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	C₂×#1343
prime radical	#23438	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

621 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 621 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57372);