#94963

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

642 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,5,8,3,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24)(25,26)(27,33,35,30)(28,31)(29,32)(34,39,36,38)(37,40);(9,11)(10,15,17,24)(12,21)(13,19)(14,20)(16,18,22,23)(25,27)(26,30)(28,36)(29,35)(31,39)(32,33)(34,37)(38,40);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)(25,28,29,37)(26,31,32,40)(27,34,35,36)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239216
maximal normal subgroup of	#72239216
maximal quotient of	#72239218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2159);