#147625

order	512
nilpotency class	4
isoclinism	isoclinic class of #13316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,23)(13,28)(14,17)(15,20)(16,27)(18,26)(21,39)(22,36)(24,34)(25,35)(29,32)(30,38)(31,40)(33,37);(9,11,14,22)(10,15,17,30)(12,24,25,40)(13,29,26,21)(16,32,33,39)(18,34,28,31)(19,35,36,23)(20,37,38,27);(2,4)(12,25)(15,30)(16,33)(20,38)(23,35)(24,40)(27,37)(31,34)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dh,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282210, #72282220, #72282356
maximal normal subgroup of	#72282210, #72282220, #72282356

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54821);