#149514

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	#346
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

161 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,29,30,49)(20,34)(21,27)(22,28)(23,25)(24,33,41,54)(26,37)(31,55,52,50)(32,51,53,71)(35,60)(36,61)(38,48)(39,44)(40,45)(42,67,59,57)(43,58,66,74)(46,63)(47,64)(56,75,73,72)(62,78)(65,69)(68,80,77,76)(70,79);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,31,21,32,35,56,22,33,36,57,38,58,62,76,23,30,37,52,39,53,63,73,40,54,64,67,65,74,79,80)(18,24,26,42,27,43,46,68,28,29,47,50,48,51,70,72,25,41,34,59,44,66,60,77,45,49,61,55,69,71,78,75);(3,9)(6,12)(10,15)(13,16)(18,25)(24,41)(26,34)(27,44)(28,45)(29,49)(42,59)(43,66)(46,60)(47,61)(48,69)(50,55)(51,71)(68,77)(70,78)(72,75)

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bh,c^a=ci,e^a=ei,c^b=cf,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231857, #72231860, #72231881, …
maximal normal subgroup of	#72231857, #72231860, #72231881, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56710);