#94814

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

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

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 132
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 132 points:

(5,6,7,14,8,15,19,39,9,16,20,40,24,43,49,79,10,17,21,41,25,44,50,80,28,46,53,82,59,85,89,114,11,18,22,42,26,45,51,81,29,47,54,83,60,86,90,115,31,48,56,84,62,87,92,116,65,88,95,117,99,118,119,130,12,13,23,34,27,35,52,69,30,36,55,70,61,73,91,104,32,37,57,71,63,74,93,105,66,76,96,107,100,110,120,125,33,38,58,72,64,75,94,106,67,77,97,108,101,111,121,126,68,78,98,109,102,112,122,127,103,113,123,128,124,129,131,132);(1,2,3,4)(6,13)(14,34)(15,35)(16,36)(17,37)(18,38)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,110)(86,111)(87,112)(88,113)(114,125)(115,126)(116,127)(117,128)(118,129)(130,132)

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=bi

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277918, #72278050
maximal normal subgroup of	#72277918, #72278050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2010);