#94505

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₄×C₃₂	C₂²
socle, prime radical	C₂²	#1397
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^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

Frattini subgroup of	#72230725, #72230729, #72230781, …
maximal subgroup of	#72230688, #72230701, #72230702, …
maximal normal subgroup of	#72230688, #72230701, #72230702, …
maximal quotient of	#72230722, #72234395, #72277893

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1701);