#94815

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₃₂	C₄²
commutator	C₄	C₄×C₃₂
Frattini, cosocle	C₂×C₆₄	C₂²
socle	C₂	#7510
prime radical	C₂²	M₁₂₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ G
LECS	G ▷ C₂×C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 128
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 128 points:

(1,2,3,10,4,12,17,38,5,13,18,39,21,44,53,84,6,14,19,40,22,45,54,85,25,47,56,86,59,90,97,119,7,15,20,36,23,46,55,78,26,48,57,79,60,91,98,113,28,49,58,80,62,92,99,114,65,93,100,115,101,121,123,127,8,11,16,37,24,41,50,81,27,42,51,82,61,87,94,116,29,43,52,83,63,88,95,117,66,89,96,118,102,120,122,128,30,9,31,32,64,33,69,72,67,34,70,73,103,75,106,109,68,35,71,74,104,76,107,110,105,77,108,111,124,112,125,126);(2,9,11,15)(3,16)(10,36,37,32)(12,33,41,46)(13,34,42,48)(14,35,43,49)(17,50)(18,51)(19,52)(20,31)(38,78,81,72)(39,79,82,73)(40,80,83,74)(44,75,87,91)(45,76,88,92)(47,77,89,93)(53,94)(54,95)(55,69)(56,96)(57,70)(58,71)(84,113,116,109)(85,114,117,110)(86,115,118,111)(90,112,120,121)(97,122)(98,106)(99,107)(100,108)(119,127,128,126)(123,125)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230722, #72230728, #72230767, …
maximal normal subgroup of	#72230722, #72230728, #72230767, …
maximal quotient of	#72277918, #72278050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2011);