#149501

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

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

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231857, #72231861, #72231862, …
maximal normal subgroup of	#72231857, #72231861, #72231862, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56697);