#94752

order	512
nilpotency class	6
isoclinism	isoclinic class of #7548
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1424
commutator	C₃₂	C₂×C₈
Frattini, cosocle	C₄×C₃₂	C₂²
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

50 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=ceh,c^2=egi,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230701, #72230767, #72230775
maximal normal subgroup of	#72230701, #72230767, #72230775
maximal quotient of	#72230767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1948);