#94063

order	512
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂³×C₁₆	C₂²
socle	C₂³	C₂²⋊C₁₆
prime radical	C₂⁴	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

224 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 76
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 76 points:

(5,6)(7,11,12,9)(8,10)(13,14,17,23,18,24,36,49,19,25,37,50,39,52,64,72,20,26,38,51,40,53,65,73,16,22,33,46,34,47,61,70)(15,27,29,54,30,55,58,74,28,44,56,48,57,68,75,71,31,45,35,62,59,69,63,76,32,21,41,42,60,43,66,67);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,32)(17,35)(18,30)(19,28)(20,31)(22,45)(23,48)(24,43)(25,27)(26,44)(29,38)(33,56)(34,60)(36,63)(37,41)(39,57)(40,59)(42,51)(46,54)(47,69)(49,71)(50,62)(52,55)(53,68)(58,65)(61,75)(64,66)(67,73)(70,74)(72,76)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,g^2=h,h^2=i,b^a=bc,c^a=cf,c^b=ci,d^b=dfi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1259);