#105680

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279621, #72279646, #72279658, …
maximal normal subgroup of	#72279621, #72279646, #72279658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12876);